phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
gastabsP (S : {set rT}) (a : aT) : a \in A -> reflect (forall x, (to x a \in S) = (x \in S)) (a \in 'N(S \| to)).	Proof. move=> Aa; apply: (iffP idP) => [nSa x\|nSa]; first exact: astabs_act. by rewrite !inE Aa; apply/subsetP=> x; rewrite inE nSa. Qed.	Lemma	gastabsP	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "aT", "apply", "astabs_act", "inE", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qact_dom_doms (H : {group rT}) : H \subset D -> qact_dom to H \subset A.	Proof. by move=> sHD; apply/subsetP=> x; rewrite qact_domE // inE; case/andP. Qed.	Lemma	qact_dom_doms	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "apply", "group", "inE", "qact_dom", "qact_domE", "sHD", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_qact_doms (H : {group rT}) : H \subset D -> [acts A, on H \| to] -> qact_dom to H :=: A.	Proof. move=> sHD aH; apply/eqP; rewrite eqEsubset; apply/andP. split; first exact: qact_dom_doms. apply/subsetP=> x Ax; rewrite qact_domE //; apply/gastabsP=> //. by move/gactsP: aH; move/(_ x Ax). Qed.	Lemma	acts_qact_doms	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "apply", "eqEsubset", "gactsP", "gastabsP", "group", "on", "qact_dom", "qact_domE", "qact_dom_doms", "sHD", "split", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qacts_cosetpre (H : {group rT}) (K' : {group coset_of H}) : H \subset D -> [acts A, on H \| to] -> [acts qact_dom to H, on K' \| to / H] -> [acts A, on coset H @*^-1 K' \| to].	Proof. move=> sHD aH aK'; apply/subsetP=> x Ax; move: (Ax) (subsetP aK'). rewrite -{1}(acts_qact_doms sHD aH) => qdx; move/(_ x qdx) => nx. rewrite !inE Ax; apply/subsetP=> y; case/morphpreP=> Ny /= K'Hy /[1!inE]. apply/morphpreP; split; first by rewrite acts_qact_dom_norm. by move/gastabsP: nx; move/(_ qdx (coset H y...	Lemma	qacts_cosetpre	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "acts_qact_dom_norm", "acts_qact_doms", "apply", "coset", "coset_of", "gastabsP", "group", "inE", "morphpreP", "on", "qactE", "qact_dom", "sHD", "split", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qacts_coset (H K : {group rT}) : H \subset D -> [acts A, on K \| to] -> [acts qact_dom to H, on (coset H) @* K \| to / H].	Proof. move=> sHD aK. apply/subsetP=> x qdx; rewrite inE qdx inE; apply/subsetP=> y. case/morphimP=> z Nz Kz /= e; rewrite e inE qactE // imset_f // inE. move/gactsP: aK; move/(_ x (subsetP (qact_dom_doms sHD) _ qdx) z); rewrite Kz. move->; move/acts_act: (acts_qact_dom to H); move/(_ x qdx z). by rewrite Nz andbT. Qed...	Lemma	qacts_coset	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "acts_act", "acts_qact_dom", "apply", "coset", "gactsP", "group", "imset_f", "inE", "morphimP", "on", "qactE", "qact_dom", "qact_dom_doms", "sHD", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxainv (B C : {set rT})	:= [max C of H \| [&& (H <\| B), ~~ (B \subset H) & [acts A, on H \| to]]].	Definition	maxainv	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "max", "on", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxainv_norm : maxainv K N -> N <\| K.	Proof. by move/maxgroupp; case/andP. Qed.	Lemma	maxainv_norm	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "maxainv", "maxgroupp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxainv_proper : maxainv K N -> N \proper K.	Proof. by move/maxgroupp; case/andP; rewrite properE; move/normal_sub->; case/andP. Qed.	Lemma	maxainv_proper	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "maxainv", "maxgroupp", "normal_sub", "proper", "properE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxainv_sub : maxainv K N -> N \subset K.	Proof. by move=> h; apply: proper_sub; apply: maxainv_proper. Qed.	Lemma	maxainv_sub	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "apply", "maxainv", "maxainv_proper", "proper_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxainv_ainvar : maxainv K N -> A \subset 'N(N \| to).	Proof. by move/maxgroupp; case/and3P. Qed.	Lemma	maxainv_ainvar	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "maxainv", "maxgroupp", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxainvS : maxainv K N -> N \subset K.	Proof. by move=> pNN; rewrite proper_sub // maxainv_proper. Qed.	Lemma	maxainvS	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "maxainv", "maxainv_proper", "proper_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxainv_exists : K :!=: 1 -> {N : {group rT} \| maxainv K N}.	Proof. move=> nt; apply: ex_maxgroup. exists [1 rT]%G. rewrite /= normal1 subG1 nt /=. apply/subsetP=> a Da; rewrite !inE Da /= sub1set !inE. by rewrite /= -actmE // morph1 eqxx. Qed.	Lemma	maxainv_exists	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "Da", "actmE", "apply", "eqxx", "ex_maxgroup", "group", "inE", "maxainv", "morph1", "normal1", "sub1set", "subG1", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxainvM (G H K : {group rT}) : H \subset D -> K \subset D -> maxainv G H -> maxainv G K -> H :<>: K -> H * K = G.	Proof. move: H K => N1 N2 sN1D sN2D pmN1 pmN2 neN12. have cN12 : commute N1 N2. apply: normC; apply: (subset_trans (maxainv_sub pmN1)). by rewrite normal_norm ?maxainv_norm. wlog nsN21 : G N1 N2 sN1D sN2D pmN1 pmN2 neN12 cN12/ ~~(N1 \subset N2). move/eqP: (neN12); rewrite eqEsubset negb_and; case/orP=> ns; first ...	Lemma	maxainvM	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "apply", "comm_joingE", "commute", "eqEsubset", "gactsM", "group", "group1", "maxainv", "maxainv_norm", "maxainv_sub", "maxgroupP", "mulG_subG", "mulg_subl", "mulg_subr", "normC", "normalM", "normal_norm", "normal_sub", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
asimple (K : {set rT})	:= maxainv K 1.	Definition	asimple	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "maxainv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
asimpleP K : reflect [/\ K :!=: 1 & forall H, H <\| K -> [acts A, on H \| to] -> H :=: 1 \/ H :=: K] (asimple K).	Proof. apply: (iffP idP). case/maxgroupP; rewrite normal1 /=; case/andP=> nsK1 aK H1. rewrite eqEsubset negb_and nsK1 /=; split => // H nHK ha. case eHK : (H :==: K); first by right; apply/eqP. left; apply: H1; rewrite ?sub1G // nHK; move/negbT: eHK. by rewrite eqEsubset negb_and normal_sub //=; move->. case=...	Lemma	asimpleP	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "Da", "actmE", "apply", "asimple", "eqEsubset", "eqxx", "inE", "maxgroupP", "morph1", "nHK", "normal1", "normal_sub", "on", "split", "sub1G", "sub1set", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acomps K s	:= ((last K s) == 1%G) && path [rel x y : {group rT} \| maxainv x y] K s.	Definition	acomps	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "group", "last", "maxainv", "path", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acompsP K s : reflect (last K s = 1%G /\ path [rel x y : {group rT} \| maxainv x y] K s) (acomps K s).	Proof. by apply: (iffP andP); case; move/eqP. Qed.	Lemma	acompsP	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "acomps", "apply", "group", "last", "maxainv", "path", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivg_acomps K s : acomps K s -> (K :==: 1) = (s == [::]).	Proof. case/andP=> ls cs; apply/eqP/eqP; last first. by move=> se; rewrite se /= in ls; apply/eqP. move=> G1; case: s ls cs => // H s _ /=; case/andP; case/maxgroupP. by rewrite G1 sub1G andbF. Qed.	Lemma	trivg_acomps	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "G1", "acomps", "apply", "last", "maxgroupP", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acomps_cons K H s : acomps K (H :: s) -> acomps H s.	Proof. by case/andP => /= ls; case/andP=> _ p; rewrite /acomps ls. Qed.	Lemma	acomps_cons	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "acomps" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
asimple_acompsP K s : acomps K s -> reflect (s = [:: 1%G]) (asimple K).	Proof. move=> cs; apply: (iffP idP); last first. by move=> se; move: cs; rewrite se /=; case/andP=> /=; rewrite andbT. case: s cs. by rewrite /acomps /= andbT; move/eqP->; case/asimpleP; rewrite eqxx. move=> H s cs sG; apply/eqP. rewrite eqseq_cons -(trivg_acomps (acomps_cons cs)) andbC andbb. case/acompsP: cs => /...	Lemma	asimple_acompsP	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "acomps", "acompsP", "acomps_cons", "apply", "asimple", "asimpleP", "eqseq_cons", "eqxx", "last", "maxainv_ainvar", "maxainv_norm", "maxainv_proper", "maxgroupP", "sG", "sub1G", "trivg_acomps" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exists_acomps K : exists s, acomps K s.	Proof. elim: {K} #\|K\| {1 3}K (leqnn #\|K\|) => [K \| n Hi K cK]. by rewrite leqNgt cardG_gt0. case/orP: (orbN (asimple K)) => [sK \| nsK]. by exists [:: (1%G : {group rT})]; rewrite /acomps eqxx /= andbT. case/orP: (orbN (K :==: 1))=> [tK \| ntK]. by exists (Nil _); rewrite /acomps /= andbT. case: (maxainv_exists ntK)...	Lemma	exists_acomps	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "Nil", "acomps", "asimple", "cardG_gt0", "eqxx", "group", "last_cons", "leqNgt", "leq_trans", "leqnn", "ltnS", "maxainv_exists", "maxainv_proper", "proper_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxainv_asimple_quo (G H : {group rT}) : H \subset D -> maxainv to G H -> asimple (to / H) (G / H).	Proof. move=> sHD /maxgroupP[/and3P[nHG pHG aH] Hmax]. apply/asimpleP; split; first by rewrite -subG1 quotient_sub1 ?normal_norm. move=> K' nK'Q aK'. have: (K' \proper (G / H)) \|\| (G / H == K'). by rewrite properE eqEsubset andbC (normal_sub nK'Q) !andbT orbC orbN. case/orP=> [ pHQ \| eQH]; last by right; apply sym_eq...	Lemma	maxainv_asimple_quo	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "apply", "asimple", "asimpleP", "coset", "cosetpre_normal", "eqEsubset", "group", "group1", "kerE", "ker_coset", "last", "maxainv", "maxgroupP", "morphimS", "morphpreK", "morphpreS", "nHG", "normal_norm", "normal_sub", "proper", "properE", "proper_subn", "qacts_cosetpre",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
asimple_quo_maxainv (G H : {group rT}) : H \subset D -> G \subset D -> [acts A, on G \| to] -> [acts A, on H \| to] -> H <\| G -> asimple (to / H) (G / H) -> maxainv to G H.	Proof. move=> sHD sGD aG aH nHG /asimpleP[ntQ maxQ]; apply/maxgroupP; split. by rewrite nHG -quotient_sub1 ?normal_norm // subG1 ntQ. move=> K /and3P[nKG nsGK aK] sHK. pose K' := (K / H)%G. have K'dQ : K' <\| (G / H)%G by apply: morphim_normal. have nKH : H <\| K by rewrite (normalS _ _ nHG) // normal_sub. have: K' :=:...	Lemma	asimple_quo_maxainv	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "aG", "acts_qact_dom_norm", "apply", "asimple", "asimpleP", "astabs_act", "group", "inE", "last", "maxainv", "maxgroupP", "morphimP", "morphim_normal", "nHG", "nKG", "nKH", "normalS", "normal_norm", "normal_refl", "normal_sub", "on", "qactE", "qact_dom", "qact_domE", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
asimpleI (N1 N2 : {group rT}) : N2 \subset 'N(N1) -> N1 \subset D -> [acts A, on N1 \| to] -> [acts A, on N2 \| to] -> asimple (to / N1) (N2 / N1) -> asimple (to / (N2 :&: N1)) (N2 / (N2 :&: N1)).	Proof. move=> nN21 sN1D aN1 aN2 /asimpleP[ntQ1 max1]. have [f1 [f1e f1ker f1pre f1im]] := restrmP (coset_morphism N1) nN21. have hf2' : N2 \subset 'N(N2 :&: N1) by apply: normsI => //; rewrite normG. have hf2'' : 'ker (coset (N2 :&: N1)) \subset 'ker f1. by rewrite f1ker !ker_coset. pose f2 := factm_morphism hf2'' h...	Lemma	asimpleI	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "apply", "asimple", "asimpleP", "card_isog", "coset", "coset_morphism", "cosetpreK", "eqEcard", "f1", "f2", "f3", "factm_morphism", "gactsI", "group", "inE", "injm_restrm", "isog", "isogP", "isog_eq1", "ker", "ker_coset", "ker_factm", "last", "leqnn", "morphim_factm",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
StrongJordanHolderUniqueness (G : {group rT}) (s1 s2 : seq {group rT}) : G \subset D -> acomps to G s1 -> acomps to G s2 -> perm_eq (mkfactors G s1) (mkfactors G s2).	Proof. have [n] := ubnP #\|G\|; elim: n G => // n Hi G in s1 s2 * => cG hsD cs1 cs2. case/orP: (orbN (G :==: 1)) => [tG \| ntG]. have -> : s1 = [::] by apply/eqP; rewrite -(trivg_acomps cs1). have -> : s2 = [::] by apply/eqP; rewrite -(trivg_acomps cs2). by rewrite /= perm_refl. case/orP: (orbN (asimple to G))=> [sG...	Lemma	StrongJordanHolderUniqueness	solvable	solvable/jordanholder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]	[ "acomps", "apply", "asimple", "asimpleI", "asimple_acompsP", "asimple_quo_maxainv", "astabs_act", "eqxx", "exists_acomps", "group", "group_inj", "inE", "isog", "isog_sym", "leq_trans", "ltnS", "maxainvM", "maxainv_asimple_quo", "maxainv_norm", "maxainv_proper", "maxainv_sub",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
charsimple A	:= [min A of G \| G :!=: 1 & G \char A].	Definition	charsimple	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "char", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frattini A	:= \bigcap_(G : {group gT} \| maximal_eq G A) G.	Definition	Frattini	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "gT", "group", "maximal_eq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frattini_group A : {group gT}	:= Eval hnf in [group of Frattini A].	Canonical	Frattini_group	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Frattini", "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting A	:= \big[dprod/1]_(p <- primes #\|A\|) 'O_p(A).	Definition	Fitting	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "dprod", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_group_set G : group_set (Fitting G).	Proof. suffices [F ->]: exists F : {group gT}, Fitting G = F by apply: groupP. rewrite /Fitting; elim: primes (primes_uniq #\|G\|) => [_\|p r IHr] /=. by exists [1 gT]%G; rewrite big_nil. case/andP=> rp /IHr[F defF]; rewrite big_cons defF. suffices{IHr} /and3P[p'F sFG nFG]: p^'.-group F && (F <\| G). have nFGp: 'O_p(G)...	Lemma	Fitting_group_set	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting", "apply", "big_cons", "big_nil", "bigdprodWY", "commG1P", "commg_subl", "commg_subr", "coprime_TIg", "dprodEY", "eq_sym", "gFnorm", "gFsub_trans", "gT", "gen0", "group", "groupP", "group_set", "inE", "joingE", "joing_idr", "norm_joinEl", "normal1", "normalM", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_group G	:= group (Fitting_group_set G).	Canonical	Fitting_group	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting_group_set", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
critical A B	:= [/\ A \char B, Frattini A \subset 'Z(A), [~: B, A] \subset 'Z(A) & 'C_B(A) = 'Z(A)].	Definition	critical	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Frattini", "char" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
special A	:= Frattini A = 'Z(A) /\ A^`(1) = 'Z(A).	Definition	special	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Frattini" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
extraspecial A	:= special A /\ prime #\|'Z(A)\|.	Definition	extraspecial	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "prime", "special" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
SCN B	:= [set A : {group gT} \| A <\| B & 'C_B(A) == A].	Definition	SCN	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
SCN_at n B	:= [set A in SCN B \| n <= 'r(A)].	Definition	SCN_at	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "SCN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Phi' ( A )"	:= (Frattini A) (format "''Phi' ( A )") : group_scope.	Notation	''Phi' ( A )	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Frattini" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Phi' ( G )"	:= (Frattini_group G) : Group_scope.	Notation	''Phi' ( G )	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Frattini_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''F' ( G )"	:= (Fitting G) (format "''F' ( G )") : group_scope.	Notation	''F' ( G )	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''F' ( G )"	:= (Fitting_group G) : Group_scope.	Notation	''F' ( G )	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''SCN' ( B )"	:= (SCN B) (format "''SCN' ( B )") : group_scope.	Notation	''SCN' ( B )	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "SCN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''SCN_' n ( B )"	:= (SCN_at n B) (n at level 2, format "''SCN_' n ( B )") : group_scope.	Notation	''SCN_' n ( B )	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "SCN_at" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pP : p.-group P.		Hypothesis	pP	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_maximal_normal : maximal M P -> M <\| P.	Proof. case/maxgroupP=> /andP[sMP sPM] maxM; rewrite /normal sMP. have:= subsetIl P 'N(M); rewrite subEproper. case/predU1P=> [/setIidPl-> // \| /maxM/= SNM]; case/negP: sPM. rewrite (nilpotent_sub_norm (pgroup_nil pP) sMP) //. by rewrite SNM // subsetI sMP normG. Qed.	Lemma	p_maximal_normal	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "maxgroupP", "maximal", "nilpotent_sub_norm", "normG", "normal", "pP", "pgroup_nil", "predU1P", "setIidPl", "subEproper", "subsetI", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_maximal_index : maximal M P -> #\|P : M\| = p.	Proof. move=> maxM; have nM := p_maximal_normal maxM. rewrite -card_quotient ?normal_norm //. rewrite -(quotient_maximal _ nM) ?normal_refl // trivg_quotient in maxM. case/maxgroupP: maxM; rewrite properEneq eq_sym sub1G andbT /=. case/(pgroup_pdiv (quotient_pgroup M pP)) => p_pr /Cauchy[] // xq. rewrite /order -cycle_...	Lemma	p_maximal_index	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Cauchy", "card_quotient", "cards1", "cycle_subG", "eq_sym", "maxgroupP", "maximal", "normal_norm", "normal_refl", "order", "pP", "p_maximal_normal", "p_pr", "pgroup_pdiv", "predU1P", "properEneq", "quotient_maximal", "quotient_pgroup", "sub1G", "subEproper", "trivg_quotient"...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_index_maximal : M \subset P -> prime #\|P : M\| -> maximal M P.	Proof. move=> sMP /primeP[lt1PM pr_PM]. apply/maxgroupP; rewrite properEcard sMP -(Lagrange sMP). rewrite -{1}(muln1 #\|M\|) ltn_pmul2l //; split=> // H sHP sMH. apply/eqP; rewrite eq_sym eqEcard sMH. case/orP: (pr_PM _ (indexSg sMH (proper_sub sHP))) => /eqP iM. by rewrite -(Lagrange sMH) iM muln1 /=. by have:= proper...	Lemma	p_index_maximal	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Lagrange", "apply", "eqEcard", "eq_sym", "indexSg", "ltn_pmul2l", "ltnn", "maxgroupP", "maximal", "muln1", "prime", "primeP", "properEcard", "proper_card", "proper_sub", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_sub G : 'Phi(G) \subset G.	Proof. by rewrite bigcap_inf // /maximal_eq eqxx. Qed.	Lemma	Phi_sub	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "bigcap_inf", "eqxx", "maximal_eq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_sub_max G M : maximal M G -> 'Phi(G) \subset M.	Proof. by move=> maxM; rewrite bigcap_inf // /maximal_eq predU1r. Qed.	Lemma	Phi_sub_max	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "bigcap_inf", "maximal", "maximal_eq", "predU1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_proper G : G :!=: 1 -> 'Phi(G) \proper G.	Proof. move/eqP; case/maximal_exists: (sub1G G) => [<- //\| [M maxM _] _]. exact: sub_proper_trans (Phi_sub_max maxM) (maxgroupp maxM). Qed.	Lemma	Phi_proper	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Phi_sub_max", "maxgroupp", "maximal_exists", "proper", "sub1G", "sub_proper_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_nongen G X : 'Phi(G) <*> X = G -> <<X>> = G.	Proof. move=> defG; have: <<X>> \subset G by rewrite -{1}defG genS ?subsetUr. case/maximal_exists=> //= [[M maxM]]; rewrite gen_subG => sXM. case/andP: (maxgroupp maxM) => _ /negP[]. by rewrite -defG gen_subG subUset Phi_sub_max. Qed.	Lemma	Phi_nongen	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Phi_sub_max", "defG", "genS", "gen_subG", "maxgroupp", "maximal_exists", "subUset", "subsetUr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frattini_continuous (rT : finGroupType) G (f : {morphism G >-> rT}) : f @* 'Phi(G) \subset 'Phi(f @* G).	Proof. apply/bigcapsP=> M maxM; rewrite sub_morphim_pre ?Phi_sub // bigcap_inf //. have {2}<-: f @^-1 (f @ G) = G by rewrite morphimGK ?subsetIl. by rewrite morphpre_maximal_eq ?maxM //; case/maximal_eqP: maxM. Qed.	Lemma	Frattini_continuous	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Phi_sub", "apply", "bigcap_inf", "bigcapsP", "maximal_eqP", "morphimGK", "morphism", "morphpre_maximal_eq", "sub_morphim_pre", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frattini_igFun	:= [igFun by Phi_sub & Frattini_continuous].	Canonical	Frattini_igFun	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Frattini_continuous", "Phi_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frattini_gFun	:= [gFun by Frattini_continuous].	Canonical	Frattini_gFun	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Frattini_continuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_char G : 'Phi(G) \char G.	Proof. exact: gFchar. Qed.	Lemma	Phi_char	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "char", "gFchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_normal G : 'Phi(G) <\| G.	Proof. exact: gFnormal. Qed.	Lemma	Phi_normal	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "gFnormal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Phi rT D G (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> f @* 'Phi(G) = 'Phi(f @* G).	Proof. exact: injmF. Qed.	Lemma	injm_Phi	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "injmF", "morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_Phi rT G (H : {group rT}) : G \isog H -> 'Phi(G) \isog 'Phi(H).	Proof. exact: gFisog. Qed.	Lemma	isog_Phi	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "gFisog", "group", "isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PhiJ G x : 'Phi(G :^ x) = 'Phi(G) :^ x.	Proof. rewrite -{1}(setIid G) -(setIidPr (Phi_sub G)) -!morphim_conj. by rewrite injm_Phi ?injm_conj. Qed.	Lemma	PhiJ	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Phi_sub", "injm_Phi", "injm_conj", "morphim_conj", "setIid", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_quotient_id G : 'Phi (G / 'Phi(G)) = 1.	Proof. apply/trivgP; rewrite -cosetpreSK cosetpre1 /=; apply/bigcapsP=> M maxM. have nPhi := Phi_normal G; have nPhiM: 'Phi(G) <\| M. by apply: normalS nPhi; [apply: bigcap_inf \| case/maximal_eqP: maxM]. by rewrite sub_cosetpre_quo ?bigcap_inf // quotient_maximal_eq. Qed.	Lemma	Phi_quotient_id	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Phi_normal", "apply", "bigcap_inf", "bigcapsP", "cosetpre1", "cosetpreSK", "maximal_eqP", "normalS", "quotient_maximal_eq", "sub_cosetpre_quo", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_quotient_cyclic G : cyclic (G / 'Phi(G)) -> cyclic G.	Proof. case/cyclicP=> /= Px; case: (cosetP Px) => x nPx ->{Px} defG. apply/cyclicP; exists x; symmetry; apply: Phi_nongen. rewrite -joing_idr norm_joinEr -?quotientK ?cycle_subG //. by rewrite /quotient morphim_cycle //= -defG quotientGK ?Phi_normal. Qed.	Lemma	Phi_quotient_cyclic	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Phi_nongen", "Phi_normal", "Px", "apply", "cosetP", "cycle_subG", "cyclic", "cyclicP", "defG", "joing_idr", "morphim_cycle", "norm_joinEr", "quotient", "quotientGK", "quotientK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivg_Phi : p.-group P -> ('Phi(P) == 1) = p.-abelem P.	Proof. move=> pP; case: (eqsVneq P 1) => [P1 \| ntP]. by rewrite P1 abelem1 -subG1 -P1 Phi_sub. have [p_pr _ _] := pgroup_pdiv pP ntP. apply/eqP/idP=> [trPhi \| abP]. apply/abelemP=> //; split=> [\|x Px]. apply/commG1P/trivgP; rewrite -trPhi. apply/bigcapsP=> M /predU1P[-> \| maxM]; first exact: der1_subG. ...	Lemma	trivg_Phi	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "P1", "Phi_sub", "Px", "TI_cardMg", "abelem", "abelem1", "abelemP", "abelem_order_p", "abelem_splits", "apply", "bigcapP", "bigcapsP", "cardSg", "card_quotient", "commG1P", "complP", "coset_idr", "cycle1", "cycle_subG", "cyclic_abelian", "der1_min", "der1_subG", "divgS", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_quotient_abelem : p.-abelem (P / 'Phi(P)).	Proof. by rewrite -trivg_Phi ?morphim_pgroup //= Phi_quotient_id. Qed.	Lemma	Phi_quotient_abelem	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Phi_quotient_id", "abelem", "morphim_pgroup", "trivg_Phi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_joing : 'Phi(P) = P^`(1) <*> 'Mho^1(P).	Proof. have [sPhiP nPhiP] := andP (Phi_normal P). apply/eqP; rewrite eqEsubset join_subG. case: (eqsVneq P 1) => [-> \| ntP] in sPhiP *. by rewrite /= (trivgP sPhiP) sub1G der_subS Mho_sub. have [p_pr _ _] := pgroup_pdiv pP ntP. have [abP x1P] := abelemP p_pr Phi_quotient_abelem. apply/andP; split. have nMP: P \subs...	Lemma	Phi_joing	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Mho", "MhoE", "Mho_sub", "Phi_normal", "Phi_quotient_abelem", "Px", "abelemP", "abelian", "apply", "coset_id", "coset_idr", "der_subS", "eqEsubset", "eqsVneq", "expn1", "gFnorm", "gFsub_trans", "gen_subG", "groupX", "imsetP", "imset_f", "join_subG", "joing_idr", "joing...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_Mho : abelian P -> 'Phi(P) = 'Mho^1(P).	Proof. by move=> cPP; rewrite Phi_joing (derG1P cPP) joing1G. Qed.	Lemma	Phi_Mho	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Mho", "Phi_joing", "abelian", "derG1P", "joing1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PhiS G H : p.-group H -> G \subset H -> 'Phi(G) \subset 'Phi(H).	Proof. move=> pH sGH; rewrite (Phi_joing pH) (Phi_joing (pgroupS sGH pH)). by rewrite genS // setUSS ?dergS ?MhoS. Qed.	Lemma	PhiS	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "MhoS", "Phi_joing", "dergS", "genS", "group", "pgroupS", "sGH", "setUSS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_Phi rT P D (f : {morphism D >-> rT}) : p.-group P -> P \subset D -> f @* 'Phi(P) = 'Phi(f @* P).	Proof. move=> pP sPD; rewrite !(@Phi_joing _ p) ?morphim_pgroup //. rewrite morphim_gen ?subUset ?gFsub_trans // morphimU -joingE. by rewrite morphimR ?morphim_Mho. Qed.	Lemma	morphim_Phi	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Phi_joing", "gFsub_trans", "group", "joingE", "morphimR", "morphimU", "morphim_Mho", "morphim_gen", "morphim_pgroup", "morphism", "pP", "subUset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_Phi P H : p.-group P -> P \subset 'N(H) -> 'Phi(P) / H = 'Phi(P / H).	Proof. exact: morphim_Phi. Qed.	Lemma	quotient_Phi	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "group", "morphim_Phi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_min G H : p.-group G -> G \subset 'N(H) -> p.-abelem (G / H) -> 'Phi(G) \subset H.	Proof. move=> pG nHG; rewrite -trivg_Phi ?quotient_pgroup // -subG1 /=. by rewrite -(quotient_Phi pG) ?quotient_sub1 // gFsub_trans. Qed.	Lemma	Phi_min	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "abelem", "gFsub_trans", "group", "nHG", "pG", "quotient_Phi", "quotient_pgroup", "quotient_sub1", "subG1", "trivg_Phi" ]	This is Aschbacher (23.2)	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_cprod G H K : p.-group G -> H \* K = G -> 'Phi(H) \* 'Phi(K) = 'Phi(G).	Proof. move=> pG defG; have [_ /mulG_sub[sHG sKG] cHK] := cprodP defG. rewrite cprodEY /=; first by rewrite (centSS (Phi_sub _) (Phi_sub _)). rewrite !(Phi_joing (pgroupS _ pG)) //=. have /cprodP[_ <- /cent_joinEr <-] := der_cprod 1 defG. have /cprodP[_ <- /cent_joinEr <-] := Mho_cprod 1 defG. by rewrite !joingA /= -!(...	Lemma	Phi_cprod	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Mho_cprod", "Phi_joing", "Phi_sub", "centSS", "cent_joinEr", "cprodEY", "cprodP", "defG", "der_cprod", "group", "joingA", "joingC", "mulG_sub", "pG", "pgroupS", "sHG", "sKG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Phi_mulg H K : p.-group H -> p.-group K -> K \subset 'C(H) -> 'Phi(H * K) = 'Phi(H) * 'Phi(K).	Proof. move=> pH pK cHK; have defHK := cprodEY cHK. have [\|_ ->] /= := cprodP (Phi_cprod _ defHK); rewrite cent_joinEr //. by rewrite pgroupM pH. Qed.	Lemma	Phi_mulg	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Phi_cprod", "cent_joinEr", "cprodEY", "cprodP", "group", "pgroupM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
charsimpleP G : reflect (G :!=: 1 /\ forall K, K :!=: 1 -> K \char G -> K :=: G) (charsimple G).	Proof. apply: (iffP mingroupP); rewrite char_refl andbT => -[ntG simG]. by split=> // K ntK chK; apply: simG; rewrite ?ntK // char_sub. by split=> // K /andP[ntK chK] _; apply: simG. Qed.	Lemma	charsimpleP	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "apply", "char", "char_refl", "char_sub", "charsimple", "mingroupP", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_normal G : 'F(G) <\| G.	Proof. rewrite -['F(G)](bigdprodWY (erefl 'F(G))). elim/big_rec: _ => [\|p H _ nsHG]; first by rewrite gen0 normal1. by rewrite -[<<_>>]joing_idr normalY ?pcore_normal. Qed.	Lemma	Fitting_normal	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "big_rec", "bigdprodWY", "gen0", "joing_idr", "normal1", "normalY", "nsHG", "pcore_normal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_sub G : 'F(G) \subset G.	Proof. by rewrite normal_sub ?Fitting_normal. Qed.	Lemma	Fitting_sub	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting_normal", "normal_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_nil G : nilpotent 'F(G).	Proof. apply: (bigdprod_nil (erefl 'F(G))) => p _. exact: pgroup_nil (pcore_pgroup p G). Qed.	Lemma	Fitting_nil	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "apply", "bigdprod_nil", "nilpotent", "pcore_pgroup", "pgroup_nil" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_max G H : H <\| G -> nilpotent H -> H \subset 'F(G).	Proof. move=> nsHG nilH; rewrite -(Sylow_gen H) gen_subG. apply/bigcupsP=> P /SylowP[p _ sylP]. case Gp: (p \in \pi(G)); last first. rewrite card1_trivg ?sub1G // (card_Hall sylP). rewrite part_p'nat // (pnat_dvd (cardSg (normal_sub nsHG))) //. by rewrite /pnat cardG_gt0 all_predC has_pred1 Gp. rewrite {P sylP}(n...	Lemma	Fitting_max	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Sub", "SylowP", "Sylow_gen", "all_predC", "apply", "big_filter", "big_mkord", "bigcup_max", "bigcupsP", "bigdprodWY", "card1_trivg", "cardG_gt0", "cardSg", "card_Hall", "dvdn_leq", "filter_pi_of", "gFnormal_trans", "gen_subG", "has_pred1", "last", "ltnS", "ltnSn", "mem_p...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_Fitting pi G : 'O_pi('F(G)) \subset 'O_pi(G).	Proof. by rewrite pcore_max ?pcore_pgroup ?gFnormal_trans ?Fitting_normal. Qed.	Lemma	pcore_Fitting	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting_normal", "gFnormal_trans", "pcore_max", "pcore_pgroup", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_core_Fitting p G : 'O_p('F(G)) = 'O_p(G).	Proof. apply/eqP; rewrite eqEsubset pcore_Fitting pcore_max ?pcore_pgroup //. apply: normalS (normal_sub (Fitting_normal _)) (pcore_normal _ _). exact: Fitting_max (pcore_normal _ _) (pgroup_nil (pcore_pgroup _ _)). Qed.	Lemma	p_core_Fitting	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting_max", "Fitting_normal", "apply", "eqEsubset", "normalS", "normal_sub", "pcore_Fitting", "pcore_max", "pcore_normal", "pcore_pgroup", "pgroup_nil" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nilpotent_Fitting G : nilpotent G -> 'F(G) = G.	Proof. by move=> nilG; apply/eqP; rewrite eqEsubset Fitting_sub Fitting_max. Qed.	Lemma	nilpotent_Fitting	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting_max", "Fitting_sub", "apply", "eqEsubset", "nilpotent" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_eq_pcore p G : 'O_p^'(G) = 1 -> 'F(G) = 'O_p(G).	Proof. move=> p'G1; have /dprodP[_ /= <- _ _] := nilpotent_pcoreC p (Fitting_nil G). by rewrite p_core_Fitting ['O_p^'(_)](trivgP _) ?mulg1 // -p'G1 pcore_Fitting. Qed.	Lemma	Fitting_eq_pcore	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting_nil", "dprodP", "mulg1", "nilpotent_pcoreC", "p_core_Fitting", "pcore_Fitting", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FittingEgen G : 'F(G) = <<\bigcup_(p < #\|G\|.+1 \| (p : nat) \in \pi(G)) 'O_p(G)>>.	Proof. apply/eqP; rewrite eqEsubset gen_subG /=. rewrite -{1}(bigdprodWY (erefl 'F(G))) (big_nth 0) big_mkord genS; last first. by apply/bigcupsP=> p _; rewrite -p_core_Fitting pcore_sub. apply/bigcupsP=> [[i /= lti]] _; set p := nth _ _ i. have pi_p: p \in \pi(G) by rewrite mem_nth. have p_dv_G: p %\| #\|G\| by rewrite...	Lemma	FittingEgen	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "apply", "big_mkord", "big_nth", "bigcup_max", "bigcupsP", "bigdprodWY", "dvdn_leq", "eqEsubset", "genS", "gen_subG", "last", "ltnS", "mem_nth", "mem_primes", "nat", "nth", "p_core_Fitting", "pcore_sub", "pi", "pi_p" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_Fitting : GFunctor.pcontinuous (@Fitting).	Proof. move=> gT rT G D f; apply: Fitting_max. by rewrite morphim_normal ?Fitting_normal. by rewrite morphim_nil ?Fitting_nil. Qed.	Lemma	morphim_Fitting	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting", "Fitting_max", "Fitting_nil", "Fitting_normal", "apply", "gT", "morphim_nil", "morphim_normal", "pcontinuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FittingS gT (G H : {group gT}) : H \subset G -> H :&: 'F(G) \subset 'F(H).	Proof. move=> sHG; rewrite -{2}(setIidPl sHG). do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; apply: morphim_Fitting. Qed.	Lemma	FittingS	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "apply", "gT", "group", "morphimIdom", "morphim_Fitting", "morphim_idm", "sHG", "setIidPl", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FittingJ gT (G : {group gT}) x : 'F(G :^ x) = 'F(G) :^ x.	Proof. rewrite !FittingEgen -genJ /= cardJg; symmetry; congr <<_>>. rewrite (big_morph (conjugate^~ x) (fun A B => conjUg A B x) (imset0 _)). by apply: eq_bigr => p _; rewrite pcoreJ. Qed.	Lemma	FittingJ	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "FittingEgen", "apply", "big_morph", "cardJg", "conjUg", "conjugate", "eq_bigr", "gT", "genJ", "group", "imset0", "pcoreJ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_igFun	:= [igFun by Fitting_sub & morphim_Fitting].	Canonical	Fitting_igFun	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Fitting_sub", "morphim_Fitting" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_gFun	:= [gFun by morphim_Fitting].	Canonical	Fitting_gFun	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "morphim_Fitting" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_pgFun	:= [pgFun by morphim_Fitting].	Canonical	Fitting_pgFun	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "morphim_Fitting" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fitting_char : 'F(G) \char G.	Proof. exact: gFchar. Qed.	Lemma	Fitting_char	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "char", "gFchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Fitting : 'injm f -> G \subset D -> f @* 'F(G) = 'F(f @* G).	Proof. exact: injmF. Qed.	Lemma	injm_Fitting	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "injmF" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_Fitting (H : {group rT}) : G \isog H -> 'F(G) \isog 'F(H).	Proof. exact: gFisog. Qed.	Lemma	isog_Fitting	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "gFisog", "group", "isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minnormal_charsimple G H : minnormal H G -> charsimple H.	Proof. case/mingroupP=> /andP[ntH nHG] minH. apply/charsimpleP; split=> // K ntK chK. by apply: minH; rewrite ?ntK (char_sub chK, char_norm_trans chK). Qed.	Lemma	minnormal_charsimple	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "apply", "char_norm_trans", "char_sub", "charsimple", "charsimpleP", "mingroupP", "minnormal", "nHG", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxnormal_charsimple G H L : G <\| L -> maxnormal H G L -> charsimple (G / H).	Proof. case/andP=> sGL nGL /maxgroupP[/andP[/andP[sHG not_sGH] nHL] maxH]. have nHG: G \subset 'N(H) := subset_trans sGL nHL. apply/charsimpleP; rewrite -subG1 quotient_sub1 //; split=> // HK ntHK chHK. case/(inv_quotientN _): (char_normal chHK) => [\|K defHK sHK]; first exact/andP. case/andP; rewrite subEproper defHK =...	Lemma	maxnormal_charsimple	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "apply", "char_norm_trans", "char_normal", "charsimple", "charsimpleP", "inv_quotientN", "maxgroupP", "maxnormal", "nHG", "nHK", "nKG", "norm_quotient_pre", "normal", "normal_norm", "predU1P", "proper_sub", "quotientGK", "quotient_norms", "quotient_sub1", "sHG", "sHK", "spl...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_split_dprod rT p (A B : {group rT}) : p.-abelem A -> B \subset A -> exists C : {group rT}, B \x C = A.	Proof. move=> abelA sBA; have [_ cAA _]:= and3P abelA. case/splitsP: (abelem_splits abelA sBA) => C /complP[tiBC defA]. by exists C; rewrite dprodE // (centSS _ sBA cAA) // -defA mulG_subr. Qed.	Lemma	abelem_split_dprod	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "abelem", "abelem_splits", "cAA", "centSS", "complP", "dprodE", "group", "mulG_subr", "splitsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_abelem_split1 rT p (A : {group rT}) x : p.-abelem A -> x \in A -> exists B : {group rT}, [/\ B \subset A, #\|B\| = #\|A\| %/ #[x] & <[x]> \x B = A].	Proof. move=> abelA Ax; have sxA: <[x]> \subset A by rewrite cycle_subG. have [B defA] := abelem_split_dprod abelA sxA. have [_ defxB _ ti_xB] := dprodP defA. have sBA: B \subset A by rewrite -defxB mulG_subr. by exists B; split; rewrite // -defxB (TI_cardMg ti_xB) mulKn ?order_gt0. Qed.	Lemma	p_abelem_split1	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "TI_cardMg", "abelem", "abelem_split_dprod", "cycle_subG", "dprodP", "group", "mulG_subr", "mulKn", "order_gt0", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_charsimple p G : p.-abelem G -> G :!=: 1 -> charsimple G.	Proof. move=> abelG ntG; apply/charsimpleP; split=> // K ntK /charP[sKG chK]. case/eqVproper: sKG => // /properP[sKG [x Gx notKx]]. have ox := abelem_order_p abelG Gx (group1_contra notKx). have [A [sAG oA defA]] := p_abelem_split1 abelG Gx. case/trivgPn: ntK => y Ky nty; have Gy := subsetP sKG y Ky. have{nty} oy := ab...	Lemma	abelem_charsimple	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "abelem", "abelemS", "abelem_order_p", "apply", "charP", "charsimple", "charsimpleP", "cycle_cyclic", "cycle_subG", "dprodP", "dprodm", "eqVproper", "eqxx", "group1_contra", "injf", "injmSK", "injm_dprodm", "isog", "isogP", "isog_abelem_card", "isog_cyclic_card", "last", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
charsimple_dprod G : charsimple G -> exists H : {group gT}, [/\ H \subset G, simple H & exists2 I : {set {perm gT}}, I \subset Aut G & \big[dprod/1]_(f in I) f @: H = G].	Proof. case/charsimpleP=> ntG simG. have [H minH sHG]: {H : {group gT} \| minnormal H G & H \subset G}. by apply: mingroup_exists; rewrite ntG normG. case/mingroupP: minH => /andP[ntH nHG] minH. pose Iok (I : {set {perm gT}}) := (I \subset Aut G) && [exists (M : {group gT} \| M <\| G), \big[dprod/1]_(f in I) f @: H ...	Lemma	charsimple_dprod	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Aut", "If", "apply", "autm", "autmE", "bigD1", "big_pred1", "bigcupP", "bigcup_max", "bigcupsP", "bigdprodWY", "card_imset", "centS", "cent_joinEr", "centsC", "cents_norm", "characteristic", "charsimple", "charsimpleP", "commG1P", "commg_subl", "commg_subr", "defG", "d...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
simple_sol_prime G : solvable G -> simple G -> prime #\|G\|.	Proof. move=> solG /simpleP[ntG simG]. have{solG} cGG: abelian G. apply/commG1P; case/simG: (der_normal 1 G) => // /eqP/idPn[]. by rewrite proper_neq // (sol_der1_proper solG). case: (trivgVpdiv G) ntG => [-> \| [p p_pr]]; first by rewrite eqxx. case/Cauchy=> // x Gx oxp _; move: p_pr; rewrite -oxp orderE. have: <[x...	Lemma	simple_sol_prime	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "Cauchy", "abelian", "apply", "cGG", "cards1", "commG1P", "cycle_subG", "der_normal", "eqxx", "orderE", "p_pr", "prime", "proper_neq", "simple", "simpleP", "sol_der1_proper", "solvable", "sub_abelian_normal", "trivgVpdiv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
charsimple_solvable G : charsimple G -> solvable G -> is_abelem G.	Proof. case/charsimple_dprod=> H [sHG simH [I Aut_I defG]] solG. have p_pr: prime #\|H\| by apply: simple_sol_prime (solvableS sHG solG) simH. set p := #\|H\| in p_pr; apply/is_abelemP; exists p => //. elim/big_rec: _ (G) defG => [_ <-\|f B If IH_B M defM]; first exact: abelem1. have [Af [[_ K _ defB] _ _ _]] := (subsetP Au...	Lemma	charsimple_solvable	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "If", "abelem1", "abelemE", "apply", "autmE", "big_rec", "charsimple", "charsimple_dprod", "cyclic_abelian", "defG", "dprodP", "dprod_abelem", "exponent_dvdn", "is_abelem", "is_abelemP", "morphimEsub", "morphim_abelem", "p_pr", "prime", "prime_cyclic", "sHG", "simple_sol_pr...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minnormal_solvable L G H : minnormal H L -> H \subset G -> solvable G -> [/\ L \subset 'N(H), H :!=: 1 & is_abelem H].	Proof. move=> minH sHG solG; have /andP[ntH nHL] := mingroupp minH. split=> //; apply: (charsimple_solvable (minnormal_charsimple minH)). exact: solvableS solG. Qed.	Lemma	minnormal_solvable	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "apply", "charsimple_solvable", "is_abelem", "mingroupp", "minnormal", "minnormal_charsimple", "sHG", "solvable", "solvableS", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
solvable_norm_abelem L G : solvable G -> G <\| L -> G :!=: 1 -> exists H : {group gT}, [/\ H \subset G, H <\| L, H :!=: 1 & is_abelem H].	Proof. move=> solG /andP[sGL nGL] ntG. have [H minH sHG]: {H : {group gT} \| minnormal H L & H \subset G}. by apply: mingroup_exists; rewrite ntG. have [nHL ntH abH] := minnormal_solvable minH sHG solG. by exists H; split; rewrite // /normal (subset_trans sHG). Qed.	Lemma	solvable_norm_abelem	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "apply", "gT", "group", "is_abelem", "mingroup_exists", "minnormal", "minnormal_solvable", "normal", "sHG", "solvable", "split", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivg_Fitting G : solvable G -> ('F(G) == 1) = (G :==: 1).	Proof. move=> solG; apply/idP/idP=> [F1 \| /eqP->]; last by rewrite gF1. apply/idPn=> /(solvable_norm_abelem solG (normal_refl _))[M [_ nsMG ntM]]. case/is_abelemP=> p _ /and3P[pM _ _]; case/negP: ntM. by rewrite -subG1 -(eqP F1) Fitting_max ?(pgroup_nil pM). Qed.	Lemma	trivg_Fitting	solvable	solvable/maximal.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...	[ "F1", "Fitting_max", "apply", "gF1", "is_abelemP", "last", "normal_refl", "pgroup_nil", "solvable", "solvable_norm_abelem", "subG1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

gastabsP (S : {set rT}) (a : aT) : a \in A -> reflect (forall x, (to x a \in S) = (x \in S)) (a \in 'N(S | to)).

Proof. move=> Aa; apply: (iffP idP) => [nSa x|nSa]; first exact: astabs_act. by rewrite !inE Aa; apply/subsetP=> x; rewrite inE nSa. Qed.

Lemma

gastabsP

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "aT", "apply", "astabs_act", "inE", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qact_dom_doms (H : {group rT}) : H \subset D -> qact_dom to H \subset A.

Proof. by move=> sHD; apply/subsetP=> x; rewrite qact_domE // inE; case/andP. Qed.

Lemma

qact_dom_doms

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "apply", "group", "inE", "qact_dom", "qact_domE", "sHD", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_qact_doms (H : {group rT}) : H \subset D -> [acts A, on H | to] -> qact_dom to H :=: A.

Proof. move=> sHD aH; apply/eqP; rewrite eqEsubset; apply/andP. split; first exact: qact_dom_doms. apply/subsetP=> x Ax; rewrite qact_domE //; apply/gastabsP=> //. by move/gactsP: aH; move/(_ x Ax). Qed.

Lemma

acts_qact_doms

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "apply", "eqEsubset", "gactsP", "gastabsP", "group", "on", "qact_dom", "qact_domE", "qact_dom_doms", "sHD", "split", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qacts_cosetpre (H : {group rT}) (K' : {group coset_of H}) : H \subset D -> [acts A, on H | to] -> [acts qact_dom to H, on K' | to / H] -> [acts A, on coset H @*^-1 K' | to].

Proof. move=> sHD aH aK'; apply/subsetP=> x Ax; move: (Ax) (subsetP aK'). rewrite -{1}(acts_qact_doms sHD aH) => qdx; move/(_ x qdx) => nx. rewrite !inE Ax; apply/subsetP=> y; case/morphpreP=> Ny /= K'Hy /[1!inE]. apply/morphpreP; split; first by rewrite acts_qact_dom_norm. by move/gastabsP: nx; move/(_ qdx (coset H y...

Lemma

qacts_cosetpre

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "acts_qact_dom_norm", "acts_qact_doms", "apply", "coset", "coset_of", "gastabsP", "group", "inE", "morphpreP", "on", "qactE", "qact_dom", "sHD", "split", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qacts_coset (H K : {group rT}) : H \subset D -> [acts A, on K | to] -> [acts qact_dom to H, on (coset H) @* K | to / H].

Proof. move=> sHD aK. apply/subsetP=> x qdx; rewrite inE qdx inE; apply/subsetP=> y. case/morphimP=> z Nz Kz /= e; rewrite e inE qactE // imset_f // inE. move/gactsP: aK; move/(_ x (subsetP (qact_dom_doms sHD) _ qdx) z); rewrite Kz. move->; move/acts_act: (acts_qact_dom to H); move/(_ x qdx z). by rewrite Nz andbT. Qed...

Lemma

qacts_coset

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "acts_act", "acts_qact_dom", "apply", "coset", "gactsP", "group", "imset_f", "inE", "morphimP", "on", "qactE", "qact_dom", "qact_dom_doms", "sHD", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxainv (B C : {set rT})

:= [max C of H | [&& (H <| B), ~~ (B \subset H) & [acts A, on H | to]]].

Definition

maxainv

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "max", "on", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxainv_norm : maxainv K N -> N <| K.

Proof. by move/maxgroupp; case/andP. Qed.

Lemma

maxainv_norm

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "maxainv", "maxgroupp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxainv_proper : maxainv K N -> N \proper K.

Proof. by move/maxgroupp; case/andP; rewrite properE; move/normal_sub->; case/andP. Qed.

Lemma

maxainv_proper

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "maxainv", "maxgroupp", "normal_sub", "proper", "properE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxainv_sub : maxainv K N -> N \subset K.

Proof. by move=> h; apply: proper_sub; apply: maxainv_proper. Qed.

Lemma

maxainv_sub

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "apply", "maxainv", "maxainv_proper", "proper_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxainv_ainvar : maxainv K N -> A \subset 'N(N | to).

Proof. by move/maxgroupp; case/and3P. Qed.

Lemma

maxainv_ainvar

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "maxainv", "maxgroupp", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxainvS : maxainv K N -> N \subset K.

Proof. by move=> pNN; rewrite proper_sub // maxainv_proper. Qed.

Lemma

maxainvS

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "maxainv", "maxainv_proper", "proper_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxainv_exists : K :!=: 1 -> {N : {group rT} | maxainv K N}.

Proof. move=> nt; apply: ex_maxgroup. exists [1 rT]%G. rewrite /= normal1 subG1 nt /=. apply/subsetP=> a Da; rewrite !inE Da /= sub1set !inE. by rewrite /= -actmE // morph1 eqxx. Qed.

Lemma

maxainv_exists

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "Da", "actmE", "apply", "eqxx", "ex_maxgroup", "group", "inE", "maxainv", "morph1", "normal1", "sub1set", "subG1", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxainvM (G H K : {group rT}) : H \subset D -> K \subset D -> maxainv G H -> maxainv G K -> H :<>: K -> H * K = G.

Proof. move: H K => N1 N2 sN1D sN2D pmN1 pmN2 neN12. have cN12 : commute N1 N2. apply: normC; apply: (subset_trans (maxainv_sub pmN1)). by rewrite normal_norm ?maxainv_norm. wlog nsN21 : G N1 N2 sN1D sN2D pmN1 pmN2 neN12 cN12/ ~~(N1 \subset N2). move/eqP: (neN12); rewrite eqEsubset negb_and; case/orP=> ns; first ...

Lemma

maxainvM

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "apply", "comm_joingE", "commute", "eqEsubset", "gactsM", "group", "group1", "maxainv", "maxainv_norm", "maxainv_sub", "maxgroupP", "mulG_subG", "mulg_subl", "mulg_subr", "normC", "normalM", "normal_norm", "normal_sub", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

asimple (K : {set rT})

:= maxainv K 1.

Definition

asimple

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "maxainv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

asimpleP K : reflect [/\ K :!=: 1 & forall H, H <| K -> [acts A, on H | to] -> H :=: 1 \/ H :=: K] (asimple K).

Proof. apply: (iffP idP). case/maxgroupP; rewrite normal1 /=; case/andP=> nsK1 aK H1. rewrite eqEsubset negb_and nsK1 /=; split => // H nHK ha. case eHK : (H :==: K); first by right; apply/eqP. left; apply: H1; rewrite ?sub1G // nHK; move/negbT: eHK. by rewrite eqEsubset negb_and normal_sub //=; move->. case=...

Lemma

asimpleP

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "Da", "actmE", "apply", "asimple", "eqEsubset", "eqxx", "inE", "maxgroupP", "morph1", "nHK", "normal1", "normal_sub", "on", "split", "sub1G", "sub1set", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acomps K s

:= ((last K s) == 1%G) && path [rel x y : {group rT} | maxainv x y] K s.

Definition

acomps

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "group", "last", "maxainv", "path", "rel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acompsP K s : reflect (last K s = 1%G /\ path [rel x y : {group rT} | maxainv x y] K s) (acomps K s).

Proof. by apply: (iffP andP); case; move/eqP. Qed.

Lemma

acompsP

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "acomps", "apply", "group", "last", "maxainv", "path", "rel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivg_acomps K s : acomps K s -> (K :==: 1) = (s == [::]).

Proof. case/andP=> ls cs; apply/eqP/eqP; last first. by move=> se; rewrite se /= in ls; apply/eqP. move=> G1; case: s ls cs => // H s _ /=; case/andP; case/maxgroupP. by rewrite G1 sub1G andbF. Qed.

Lemma

trivg_acomps

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "G1", "acomps", "apply", "last", "maxgroupP", "sub1G" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acomps_cons K H s : acomps K (H :: s) -> acomps H s.

Proof. by case/andP => /= ls; case/andP=> _ p; rewrite /acomps ls. Qed.

Lemma

acomps_cons

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "acomps" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

asimple_acompsP K s : acomps K s -> reflect (s = [:: 1%G]) (asimple K).

Proof. move=> cs; apply: (iffP idP); last first. by move=> se; move: cs; rewrite se /=; case/andP=> /=; rewrite andbT. case: s cs. by rewrite /acomps /= andbT; move/eqP->; case/asimpleP; rewrite eqxx. move=> H s cs sG; apply/eqP. rewrite eqseq_cons -(trivg_acomps (acomps_cons cs)) andbC andbb. case/acompsP: cs => /...

Lemma

asimple_acompsP

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "acomps", "acompsP", "acomps_cons", "apply", "asimple", "asimpleP", "eqseq_cons", "eqxx", "last", "maxainv_ainvar", "maxainv_norm", "maxainv_proper", "maxgroupP", "sG", "sub1G", "trivg_acomps" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exists_acomps K : exists s, acomps K s.

Proof. elim: {K} #|K| {1 3}K (leqnn #|K|) => [K | n Hi K cK]. by rewrite leqNgt cardG_gt0. case/orP: (orbN (asimple K)) => [sK | nsK]. by exists [:: (1%G : {group rT})]; rewrite /acomps eqxx /= andbT. case/orP: (orbN (K :==: 1))=> [tK | ntK]. by exists (Nil _); rewrite /acomps /= andbT. case: (maxainv_exists ntK)...

Lemma

exists_acomps

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "Nil", "acomps", "asimple", "cardG_gt0", "eqxx", "group", "last_cons", "leqNgt", "leq_trans", "leqnn", "ltnS", "maxainv_exists", "maxainv_proper", "proper_card" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxainv_asimple_quo (G H : {group rT}) : H \subset D -> maxainv to G H -> asimple (to / H) (G / H).

Proof. move=> sHD /maxgroupP[/and3P[nHG pHG aH] Hmax]. apply/asimpleP; split; first by rewrite -subG1 quotient_sub1 ?normal_norm. move=> K' nK'Q aK'. have: (K' \proper (G / H)) || (G / H == K'). by rewrite properE eqEsubset andbC (normal_sub nK'Q) !andbT orbC orbN. case/orP=> [ pHQ | eQH]; last by right; apply sym_eq...

Lemma

maxainv_asimple_quo

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "apply", "asimple", "asimpleP", "coset", "cosetpre_normal", "eqEsubset", "group", "group1", "kerE", "ker_coset", "last", "maxainv", "maxgroupP", "morphimS", "morphpreK", "morphpreS", "nHG", "normal_norm", "normal_sub", "proper", "properE", "proper_subn", "qacts_cosetpre",...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

asimple_quo_maxainv (G H : {group rT}) : H \subset D -> G \subset D -> [acts A, on G | to] -> [acts A, on H | to] -> H <| G -> asimple (to / H) (G / H) -> maxainv to G H.

Proof. move=> sHD sGD aG aH nHG /asimpleP[ntQ maxQ]; apply/maxgroupP; split. by rewrite nHG -quotient_sub1 ?normal_norm // subG1 ntQ. move=> K /and3P[nKG nsGK aK] sHK. pose K' := (K / H)%G. have K'dQ : K' <| (G / H)%G by apply: morphim_normal. have nKH : H <| K by rewrite (normalS _ _ nHG) // normal_sub. have: K' :=:...

Lemma

asimple_quo_maxainv

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "aG", "acts_qact_dom_norm", "apply", "asimple", "asimpleP", "astabs_act", "group", "inE", "last", "maxainv", "maxgroupP", "morphimP", "morphim_normal", "nHG", "nKG", "nKH", "normalS", "normal_norm", "normal_refl", "normal_sub", "on", "qactE", "qact_dom", "qact_domE", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

asimpleI (N1 N2 : {group rT}) : N2 \subset 'N(N1) -> N1 \subset D -> [acts A, on N1 | to] -> [acts A, on N2 | to] -> asimple (to / N1) (N2 / N1) -> asimple (to / (N2 :&: N1)) (N2 / (N2 :&: N1)).

Proof. move=> nN21 sN1D aN1 aN2 /asimpleP[ntQ1 max1]. have [f1 [f1e f1ker f1pre f1im]] := restrmP (coset_morphism N1) nN21. have hf2' : N2 \subset 'N(N2 :&: N1) by apply: normsI => //; rewrite normG. have hf2'' : 'ker (coset (N2 :&: N1)) \subset 'ker f1. by rewrite f1ker !ker_coset. pose f2 := factm_morphism hf2'' h...

Lemma

asimpleI

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "apply", "asimple", "asimpleP", "card_isog", "coset", "coset_morphism", "cosetpreK", "eqEcard", "f1", "f2", "f3", "factm_morphism", "gactsI", "group", "inE", "injm_restrm", "isog", "isogP", "isog_eq1", "ker", "ker_coset", "ker_factm", "last", "leqnn", "morphim_factm",...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

StrongJordanHolderUniqueness (G : {group rT}) (s1 s2 : seq {group rT}) : G \subset D -> acomps to G s1 -> acomps to G s2 -> perm_eq (mkfactors G s1) (mkfactors G s2).

Proof. have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => cG hsD cs1 cs2. case/orP: (orbN (G :==: 1)) => [tG | ntG]. have -> : s1 = [::] by apply/eqP; rewrite -(trivg_acomps cs1). have -> : s2 = [::] by apply/eqP; rewrite -(trivg_acomps cs2). by rewrite /= perm_refl. case/orP: (orbN (asimple to G))=> [sG...

Lemma

StrongJordanHolderUniqueness

solvable

solvable/jordanholder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "gseries" ]

[ "acomps", "apply", "asimple", "asimpleI", "asimple_acompsP", "asimple_quo_maxainv", "astabs_act", "eqxx", "exists_acomps", "group", "group_inj", "inE", "isog", "isog_sym", "leq_trans", "ltnS", "maxainvM", "maxainv_asimple_quo", "maxainv_norm", "maxainv_proper", "maxainv_sub",...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

charsimple A

:= [min A of G | G :!=: 1 & G \char A].

Definition

charsimple

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "char", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frattini A

:= \bigcap_(G : {group gT} | maximal_eq G A) G.

Definition

Frattini

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "gT", "group", "maximal_eq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frattini_group A : {group gT}

:= Eval hnf in [group of Frattini A].

Canonical

Frattini_group

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Frattini", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting A

:= \big[dprod/1]_(p <- primes #|A|) 'O_p(A).

Definition

Fitting

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "dprod", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_group_set G : group_set (Fitting G).

Proof. suffices [F ->]: exists F : {group gT}, Fitting G = F by apply: groupP. rewrite /Fitting; elim: primes (primes_uniq #|G|) => [_|p r IHr] /=. by exists [1 gT]%G; rewrite big_nil. case/andP=> rp /IHr[F defF]; rewrite big_cons defF. suffices{IHr} /and3P[p'F sFG nFG]: p^'.-group F && (F <| G). have nFGp: 'O_p(G)...

Lemma

Fitting_group_set

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting", "apply", "big_cons", "big_nil", "bigdprodWY", "commG1P", "commg_subl", "commg_subr", "coprime_TIg", "dprodEY", "eq_sym", "gFnorm", "gFsub_trans", "gT", "gen0", "group", "groupP", "group_set", "inE", "joingE", "joing_idr", "norm_joinEl", "normal1", "normalM", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_group G

:= group (Fitting_group_set G).

Canonical

Fitting_group

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting_group_set", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

critical A B

:= [/\ A \char B, Frattini A \subset 'Z(A), [~: B, A] \subset 'Z(A) & 'C_B(A) = 'Z(A)].

Definition

critical

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Frattini", "char" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

special A

:= Frattini A = 'Z(A) /\ A^`(1) = 'Z(A).

Definition

special

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Frattini" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

extraspecial A

:= special A /\ prime #|'Z(A)|.

Definition

extraspecial

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "prime", "special" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

SCN B

:= [set A : {group gT} | A <| B & 'C_B(A) == A].

Definition

SCN

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

SCN_at n B

:= [set A in SCN B | n <= 'r(A)].

Definition

SCN_at

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "SCN" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Phi' ( A )"

:= (Frattini A) (format "''Phi' ( A )") : group_scope.

Notation

''Phi' ( A )

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Frattini" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Phi' ( G )"

:= (Frattini_group G) : Group_scope.

Notation

''Phi' ( G )

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Frattini_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''F' ( G )"

:= (Fitting G) (format "''F' ( G )") : group_scope.

Notation

''F' ( G )

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''F' ( G )"

:= (Fitting_group G) : Group_scope.

Notation

''F' ( G )

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''SCN' ( B )"

:= (SCN B) (format "''SCN' ( B )") : group_scope.

Notation

''SCN' ( B )

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "SCN" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''SCN_' n ( B )"

:= (SCN_at n B) (n at level 2, format "''SCN_' n ( B )") : group_scope.

Notation

''SCN_' n ( B )

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "SCN_at" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pP : p.-group P.

Hypothesis

pP

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_maximal_normal : maximal M P -> M <| P.

Proof. case/maxgroupP=> /andP[sMP sPM] maxM; rewrite /normal sMP. have:= subsetIl P 'N(M); rewrite subEproper. case/predU1P=> [/setIidPl-> // | /maxM/= SNM]; case/negP: sPM. rewrite (nilpotent_sub_norm (pgroup_nil pP) sMP) //. by rewrite SNM // subsetI sMP normG. Qed.

Lemma

p_maximal_normal

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "maxgroupP", "maximal", "nilpotent_sub_norm", "normG", "normal", "pP", "pgroup_nil", "predU1P", "setIidPl", "subEproper", "subsetI", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_maximal_index : maximal M P -> #|P : M| = p.

Proof. move=> maxM; have nM := p_maximal_normal maxM. rewrite -card_quotient ?normal_norm //. rewrite -(quotient_maximal _ nM) ?normal_refl // trivg_quotient in maxM. case/maxgroupP: maxM; rewrite properEneq eq_sym sub1G andbT /=. case/(pgroup_pdiv (quotient_pgroup M pP)) => p_pr /Cauchy[] // xq. rewrite /order -cycle_...

Lemma

p_maximal_index

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Cauchy", "card_quotient", "cards1", "cycle_subG", "eq_sym", "maxgroupP", "maximal", "normal_norm", "normal_refl", "order", "pP", "p_maximal_normal", "p_pr", "pgroup_pdiv", "predU1P", "properEneq", "quotient_maximal", "quotient_pgroup", "sub1G", "subEproper", "trivg_quotient"...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_index_maximal : M \subset P -> prime #|P : M| -> maximal M P.

Proof. move=> sMP /primeP[lt1PM pr_PM]. apply/maxgroupP; rewrite properEcard sMP -(Lagrange sMP). rewrite -{1}(muln1 #|M|) ltn_pmul2l //; split=> // H sHP sMH. apply/eqP; rewrite eq_sym eqEcard sMH. case/orP: (pr_PM _ (indexSg sMH (proper_sub sHP))) => /eqP iM. by rewrite -(Lagrange sMH) iM muln1 /=. by have:= proper...

Lemma

p_index_maximal

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Lagrange", "apply", "eqEcard", "eq_sym", "indexSg", "ltn_pmul2l", "ltnn", "maxgroupP", "maximal", "muln1", "prime", "primeP", "properEcard", "proper_card", "proper_sub", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_sub G : 'Phi(G) \subset G.

Proof. by rewrite bigcap_inf // /maximal_eq eqxx. Qed.

Lemma

Phi_sub

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "bigcap_inf", "eqxx", "maximal_eq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_sub_max G M : maximal M G -> 'Phi(G) \subset M.

Proof. by move=> maxM; rewrite bigcap_inf // /maximal_eq predU1r. Qed.

Lemma

Phi_sub_max

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "bigcap_inf", "maximal", "maximal_eq", "predU1r" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_proper G : G :!=: 1 -> 'Phi(G) \proper G.

Proof. move/eqP; case/maximal_exists: (sub1G G) => [<- //| [M maxM _] _]. exact: sub_proper_trans (Phi_sub_max maxM) (maxgroupp maxM). Qed.

Lemma

Phi_proper

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Phi_sub_max", "maxgroupp", "maximal_exists", "proper", "sub1G", "sub_proper_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_nongen G X : 'Phi(G) <*> X = G -> <<X>> = G.

Proof. move=> defG; have: <<X>> \subset G by rewrite -{1}defG genS ?subsetUr. case/maximal_exists=> //= [[M maxM]]; rewrite gen_subG => sXM. case/andP: (maxgroupp maxM) => _ /negP[]. by rewrite -defG gen_subG subUset Phi_sub_max. Qed.

Lemma

Phi_nongen

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Phi_sub_max", "defG", "genS", "gen_subG", "maxgroupp", "maximal_exists", "subUset", "subsetUr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frattini_continuous (rT : finGroupType) G (f : {morphism G >-> rT}) : f @* 'Phi(G) \subset 'Phi(f @* G).

Proof. apply/bigcapsP=> M maxM; rewrite sub_morphim_pre ?Phi_sub // bigcap_inf //. have {2}<-: f @*^-1 (f @* G) = G by rewrite morphimGK ?subsetIl. by rewrite morphpre_maximal_eq ?maxM //; case/maximal_eqP: maxM. Qed.

Lemma

Frattini_continuous

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Phi_sub", "apply", "bigcap_inf", "bigcapsP", "maximal_eqP", "morphimGK", "morphism", "morphpre_maximal_eq", "sub_morphim_pre", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frattini_igFun

:= [igFun by Phi_sub & Frattini_continuous].

Canonical

Frattini_igFun

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Frattini_continuous", "Phi_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frattini_gFun

:= [gFun by Frattini_continuous].

Canonical

Frattini_gFun

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Frattini_continuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_char G : 'Phi(G) \char G.

Proof. exact: gFchar. Qed.

Lemma

Phi_char

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "char", "gFchar" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_normal G : 'Phi(G) <| G.

Proof. exact: gFnormal. Qed.

Lemma

Phi_normal

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "gFnormal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Phi rT D G (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> f @* 'Phi(G) = 'Phi(f @* G).

Proof. exact: injmF. Qed.

Lemma

injm_Phi

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "injmF", "morphism" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_Phi rT G (H : {group rT}) : G \isog H -> 'Phi(G) \isog 'Phi(H).

Proof. exact: gFisog. Qed.

Lemma

isog_Phi

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "gFisog", "group", "isog" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PhiJ G x : 'Phi(G :^ x) = 'Phi(G) :^ x.

Proof. rewrite -{1}(setIid G) -(setIidPr (Phi_sub G)) -!morphim_conj. by rewrite injm_Phi ?injm_conj. Qed.

Lemma

PhiJ

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Phi_sub", "injm_Phi", "injm_conj", "morphim_conj", "setIid", "setIidPr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_quotient_id G : 'Phi (G / 'Phi(G)) = 1.

Proof. apply/trivgP; rewrite -cosetpreSK cosetpre1 /=; apply/bigcapsP=> M maxM. have nPhi := Phi_normal G; have nPhiM: 'Phi(G) <| M. by apply: normalS nPhi; [apply: bigcap_inf | case/maximal_eqP: maxM]. by rewrite sub_cosetpre_quo ?bigcap_inf // quotient_maximal_eq. Qed.

Lemma

Phi_quotient_id

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Phi_normal", "apply", "bigcap_inf", "bigcapsP", "cosetpre1", "cosetpreSK", "maximal_eqP", "normalS", "quotient_maximal_eq", "sub_cosetpre_quo", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_quotient_cyclic G : cyclic (G / 'Phi(G)) -> cyclic G.

Proof. case/cyclicP=> /= Px; case: (cosetP Px) => x nPx ->{Px} defG. apply/cyclicP; exists x; symmetry; apply: Phi_nongen. rewrite -joing_idr norm_joinEr -?quotientK ?cycle_subG //. by rewrite /quotient morphim_cycle //= -defG quotientGK ?Phi_normal. Qed.

Lemma

Phi_quotient_cyclic

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Phi_nongen", "Phi_normal", "Px", "apply", "cosetP", "cycle_subG", "cyclic", "cyclicP", "defG", "joing_idr", "morphim_cycle", "norm_joinEr", "quotient", "quotientGK", "quotientK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivg_Phi : p.-group P -> ('Phi(P) == 1) = p.-abelem P.

Proof. move=> pP; case: (eqsVneq P 1) => [P1 | ntP]. by rewrite P1 abelem1 -subG1 -P1 Phi_sub. have [p_pr _ _] := pgroup_pdiv pP ntP. apply/eqP/idP=> [trPhi | abP]. apply/abelemP=> //; split=> [|x Px]. apply/commG1P/trivgP; rewrite -trPhi. apply/bigcapsP=> M /predU1P[-> | maxM]; first exact: der1_subG. ...

Lemma

trivg_Phi

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "P1", "Phi_sub", "Px", "TI_cardMg", "abelem", "abelem1", "abelemP", "abelem_order_p", "abelem_splits", "apply", "bigcapP", "bigcapsP", "cardSg", "card_quotient", "commG1P", "complP", "coset_idr", "cycle1", "cycle_subG", "cyclic_abelian", "der1_min", "der1_subG", "divgS", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_quotient_abelem : p.-abelem (P / 'Phi(P)).

Proof. by rewrite -trivg_Phi ?morphim_pgroup //= Phi_quotient_id. Qed.

Lemma

Phi_quotient_abelem

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Phi_quotient_id", "abelem", "morphim_pgroup", "trivg_Phi" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_joing : 'Phi(P) = P^`(1) <*> 'Mho^1(P).

Proof. have [sPhiP nPhiP] := andP (Phi_normal P). apply/eqP; rewrite eqEsubset join_subG. case: (eqsVneq P 1) => [-> | ntP] in sPhiP *. by rewrite /= (trivgP sPhiP) sub1G der_subS Mho_sub. have [p_pr _ _] := pgroup_pdiv pP ntP. have [abP x1P] := abelemP p_pr Phi_quotient_abelem. apply/andP; split. have nMP: P \subs...

Lemma

Phi_joing

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Mho", "MhoE", "Mho_sub", "Phi_normal", "Phi_quotient_abelem", "Px", "abelemP", "abelian", "apply", "coset_id", "coset_idr", "der_subS", "eqEsubset", "eqsVneq", "expn1", "gFnorm", "gFsub_trans", "gen_subG", "groupX", "imsetP", "imset_f", "join_subG", "joing_idr", "joing...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_Mho : abelian P -> 'Phi(P) = 'Mho^1(P).

Proof. by move=> cPP; rewrite Phi_joing (derG1P cPP) joing1G. Qed.

Lemma

Phi_Mho

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Mho", "Phi_joing", "abelian", "derG1P", "joing1G" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PhiS G H : p.-group H -> G \subset H -> 'Phi(G) \subset 'Phi(H).

Proof. move=> pH sGH; rewrite (Phi_joing pH) (Phi_joing (pgroupS sGH pH)). by rewrite genS // setUSS ?dergS ?MhoS. Qed.

Lemma

PhiS

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "MhoS", "Phi_joing", "dergS", "genS", "group", "pgroupS", "sGH", "setUSS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_Phi rT P D (f : {morphism D >-> rT}) : p.-group P -> P \subset D -> f @* 'Phi(P) = 'Phi(f @* P).

Proof. move=> pP sPD; rewrite !(@Phi_joing _ p) ?morphim_pgroup //. rewrite morphim_gen ?subUset ?gFsub_trans // morphimU -joingE. by rewrite morphimR ?morphim_Mho. Qed.

Lemma

morphim_Phi

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Phi_joing", "gFsub_trans", "group", "joingE", "morphimR", "morphimU", "morphim_Mho", "morphim_gen", "morphim_pgroup", "morphism", "pP", "subUset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_Phi P H : p.-group P -> P \subset 'N(H) -> 'Phi(P) / H = 'Phi(P / H).

Proof. exact: morphim_Phi. Qed.

Lemma

quotient_Phi

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "group", "morphim_Phi" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_min G H : p.-group G -> G \subset 'N(H) -> p.-abelem (G / H) -> 'Phi(G) \subset H.

Proof. move=> pG nHG; rewrite -trivg_Phi ?quotient_pgroup // -subG1 /=. by rewrite -(quotient_Phi pG) ?quotient_sub1 // gFsub_trans. Qed.

Lemma

Phi_min

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "abelem", "gFsub_trans", "group", "nHG", "pG", "quotient_Phi", "quotient_pgroup", "quotient_sub1", "subG1", "trivg_Phi" ]

This is Aschbacher (23.2)

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_cprod G H K : p.-group G -> H \* K = G -> 'Phi(H) \* 'Phi(K) = 'Phi(G).

Proof. move=> pG defG; have [_ /mulG_sub[sHG sKG] cHK] := cprodP defG. rewrite cprodEY /=; first by rewrite (centSS (Phi_sub _) (Phi_sub _)). rewrite !(Phi_joing (pgroupS _ pG)) //=. have /cprodP[_ <- /cent_joinEr <-] := der_cprod 1 defG. have /cprodP[_ <- /cent_joinEr <-] := Mho_cprod 1 defG. by rewrite !joingA /= -!(...

Lemma

Phi_cprod

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Mho_cprod", "Phi_joing", "Phi_sub", "centSS", "cent_joinEr", "cprodEY", "cprodP", "defG", "der_cprod", "group", "joingA", "joingC", "mulG_sub", "pG", "pgroupS", "sHG", "sKG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Phi_mulg H K : p.-group H -> p.-group K -> K \subset 'C(H) -> 'Phi(H * K) = 'Phi(H) * 'Phi(K).

Proof. move=> pH pK cHK; have defHK := cprodEY cHK. have [|_ ->] /= := cprodP (Phi_cprod _ defHK); rewrite cent_joinEr //. by rewrite pgroupM pH. Qed.

Lemma

Phi_mulg

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Phi_cprod", "cent_joinEr", "cprodEY", "cprodP", "group", "pgroupM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

charsimpleP G : reflect (G :!=: 1 /\ forall K, K :!=: 1 -> K \char G -> K :=: G) (charsimple G).

Proof. apply: (iffP mingroupP); rewrite char_refl andbT => -[ntG simG]. by split=> // K ntK chK; apply: simG; rewrite ?ntK // char_sub. by split=> // K /andP[ntK chK] _; apply: simG. Qed.

Lemma

charsimpleP

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "apply", "char", "char_refl", "char_sub", "charsimple", "mingroupP", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_normal G : 'F(G) <| G.

Proof. rewrite -['F(G)](bigdprodWY (erefl 'F(G))). elim/big_rec: _ => [|p H _ nsHG]; first by rewrite gen0 normal1. by rewrite -[<<_>>]joing_idr normalY ?pcore_normal. Qed.

Lemma

Fitting_normal

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "big_rec", "bigdprodWY", "gen0", "joing_idr", "normal1", "normalY", "nsHG", "pcore_normal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_sub G : 'F(G) \subset G.

Proof. by rewrite normal_sub ?Fitting_normal. Qed.

Lemma

Fitting_sub

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting_normal", "normal_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_nil G : nilpotent 'F(G).

Proof. apply: (bigdprod_nil (erefl 'F(G))) => p _. exact: pgroup_nil (pcore_pgroup p G). Qed.

Lemma

Fitting_nil

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "apply", "bigdprod_nil", "nilpotent", "pcore_pgroup", "pgroup_nil" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_max G H : H <| G -> nilpotent H -> H \subset 'F(G).

Proof. move=> nsHG nilH; rewrite -(Sylow_gen H) gen_subG. apply/bigcupsP=> P /SylowP[p _ sylP]. case Gp: (p \in \pi(G)); last first. rewrite card1_trivg ?sub1G // (card_Hall sylP). rewrite part_p'nat // (pnat_dvd (cardSg (normal_sub nsHG))) //. by rewrite /pnat cardG_gt0 all_predC has_pred1 Gp. rewrite {P sylP}(n...

Lemma

Fitting_max

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Sub", "SylowP", "Sylow_gen", "all_predC", "apply", "big_filter", "big_mkord", "bigcup_max", "bigcupsP", "bigdprodWY", "card1_trivg", "cardG_gt0", "cardSg", "card_Hall", "dvdn_leq", "filter_pi_of", "gFnormal_trans", "gen_subG", "has_pred1", "last", "ltnS", "ltnSn", "mem_p...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_Fitting pi G : 'O_pi('F(G)) \subset 'O_pi(G).

Proof. by rewrite pcore_max ?pcore_pgroup ?gFnormal_trans ?Fitting_normal. Qed.

Lemma

pcore_Fitting

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting_normal", "gFnormal_trans", "pcore_max", "pcore_pgroup", "pi" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_core_Fitting p G : 'O_p('F(G)) = 'O_p(G).

Proof. apply/eqP; rewrite eqEsubset pcore_Fitting pcore_max ?pcore_pgroup //. apply: normalS (normal_sub (Fitting_normal _)) (pcore_normal _ _). exact: Fitting_max (pcore_normal _ _) (pgroup_nil (pcore_pgroup _ _)). Qed.

Lemma

p_core_Fitting

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting_max", "Fitting_normal", "apply", "eqEsubset", "normalS", "normal_sub", "pcore_Fitting", "pcore_max", "pcore_normal", "pcore_pgroup", "pgroup_nil" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nilpotent_Fitting G : nilpotent G -> 'F(G) = G.

Proof. by move=> nilG; apply/eqP; rewrite eqEsubset Fitting_sub Fitting_max. Qed.

Lemma

nilpotent_Fitting

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting_max", "Fitting_sub", "apply", "eqEsubset", "nilpotent" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_eq_pcore p G : 'O_p^'(G) = 1 -> 'F(G) = 'O_p(G).

Proof. move=> p'G1; have /dprodP[_ /= <- _ _] := nilpotent_pcoreC p (Fitting_nil G). by rewrite p_core_Fitting ['O_p^'(_)](trivgP _) ?mulg1 // -p'G1 pcore_Fitting. Qed.

Lemma

Fitting_eq_pcore

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting_nil", "dprodP", "mulg1", "nilpotent_pcoreC", "p_core_Fitting", "pcore_Fitting", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FittingEgen G : 'F(G) = <<\bigcup_(p < #|G|.+1 | (p : nat) \in \pi(G)) 'O_p(G)>>.

Proof. apply/eqP; rewrite eqEsubset gen_subG /=. rewrite -{1}(bigdprodWY (erefl 'F(G))) (big_nth 0) big_mkord genS; last first. by apply/bigcupsP=> p _; rewrite -p_core_Fitting pcore_sub. apply/bigcupsP=> [[i /= lti]] _; set p := nth _ _ i. have pi_p: p \in \pi(G) by rewrite mem_nth. have p_dv_G: p %| #|G| by rewrite...

Lemma

FittingEgen

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "apply", "big_mkord", "big_nth", "bigcup_max", "bigcupsP", "bigdprodWY", "dvdn_leq", "eqEsubset", "genS", "gen_subG", "last", "ltnS", "mem_nth", "mem_primes", "nat", "nth", "p_core_Fitting", "pcore_sub", "pi", "pi_p" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_Fitting : GFunctor.pcontinuous (@Fitting).

Proof. move=> gT rT G D f; apply: Fitting_max. by rewrite morphim_normal ?Fitting_normal. by rewrite morphim_nil ?Fitting_nil. Qed.

Lemma

morphim_Fitting

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting", "Fitting_max", "Fitting_nil", "Fitting_normal", "apply", "gT", "morphim_nil", "morphim_normal", "pcontinuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FittingS gT (G H : {group gT}) : H \subset G -> H :&: 'F(G) \subset 'F(H).

Proof. move=> sHG; rewrite -{2}(setIidPl sHG). do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; apply: morphim_Fitting. Qed.

Lemma

FittingS

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "apply", "gT", "group", "morphimIdom", "morphim_Fitting", "morphim_idm", "sHG", "setIidPl", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FittingJ gT (G : {group gT}) x : 'F(G :^ x) = 'F(G) :^ x.

Proof. rewrite !FittingEgen -genJ /= cardJg; symmetry; congr <<_>>. rewrite (big_morph (conjugate^~ x) (fun A B => conjUg A B x) (imset0 _)). by apply: eq_bigr => p _; rewrite pcoreJ. Qed.

Lemma

FittingJ

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "FittingEgen", "apply", "big_morph", "cardJg", "conjUg", "conjugate", "eq_bigr", "gT", "genJ", "group", "imset0", "pcoreJ" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_igFun

:= [igFun by Fitting_sub & morphim_Fitting].

Canonical

Fitting_igFun

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Fitting_sub", "morphim_Fitting" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_gFun

:= [gFun by morphim_Fitting].

Canonical

Fitting_gFun

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "morphim_Fitting" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_pgFun

:= [pgFun by morphim_Fitting].

Canonical

Fitting_pgFun

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "morphim_Fitting" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fitting_char : 'F(G) \char G.

Proof. exact: gFchar. Qed.

Lemma

Fitting_char

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "char", "gFchar" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Fitting : 'injm f -> G \subset D -> f @* 'F(G) = 'F(f @* G).

Proof. exact: injmF. Qed.

Lemma

injm_Fitting

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "injmF" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_Fitting (H : {group rT}) : G \isog H -> 'F(G) \isog 'F(H).

Proof. exact: gFisog. Qed.

Lemma

isog_Fitting

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "gFisog", "group", "isog" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minnormal_charsimple G H : minnormal H G -> charsimple H.

Proof. case/mingroupP=> /andP[ntH nHG] minH. apply/charsimpleP; split=> // K ntK chK. by apply: minH; rewrite ?ntK (char_sub chK, char_norm_trans chK). Qed.

Lemma

minnormal_charsimple

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "apply", "char_norm_trans", "char_sub", "charsimple", "charsimpleP", "mingroupP", "minnormal", "nHG", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxnormal_charsimple G H L : G <| L -> maxnormal H G L -> charsimple (G / H).

Proof. case/andP=> sGL nGL /maxgroupP[/andP[/andP[sHG not_sGH] nHL] maxH]. have nHG: G \subset 'N(H) := subset_trans sGL nHL. apply/charsimpleP; rewrite -subG1 quotient_sub1 //; split=> // HK ntHK chHK. case/(inv_quotientN _): (char_normal chHK) => [|K defHK sHK]; first exact/andP. case/andP; rewrite subEproper defHK =...

Lemma

maxnormal_charsimple

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "apply", "char_norm_trans", "char_normal", "charsimple", "charsimpleP", "inv_quotientN", "maxgroupP", "maxnormal", "nHG", "nHK", "nKG", "norm_quotient_pre", "normal", "normal_norm", "predU1P", "proper_sub", "quotientGK", "quotient_norms", "quotient_sub1", "sHG", "sHK", "spl...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_split_dprod rT p (A B : {group rT}) : p.-abelem A -> B \subset A -> exists C : {group rT}, B \x C = A.

Proof. move=> abelA sBA; have [_ cAA _]:= and3P abelA. case/splitsP: (abelem_splits abelA sBA) => C /complP[tiBC defA]. by exists C; rewrite dprodE // (centSS _ sBA cAA) // -defA mulG_subr. Qed.

Lemma

abelem_split_dprod

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "abelem", "abelem_splits", "cAA", "centSS", "complP", "dprodE", "group", "mulG_subr", "splitsP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_abelem_split1 rT p (A : {group rT}) x : p.-abelem A -> x \in A -> exists B : {group rT}, [/\ B \subset A, #|B| = #|A| %/ #[x] & <[x]> \x B = A].

Proof. move=> abelA Ax; have sxA: <[x]> \subset A by rewrite cycle_subG. have [B defA] := abelem_split_dprod abelA sxA. have [_ defxB _ ti_xB] := dprodP defA. have sBA: B \subset A by rewrite -defxB mulG_subr. by exists B; split; rewrite // -defxB (TI_cardMg ti_xB) mulKn ?order_gt0. Qed.

Lemma

p_abelem_split1

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "TI_cardMg", "abelem", "abelem_split_dprod", "cycle_subG", "dprodP", "group", "mulG_subr", "mulKn", "order_gt0", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_charsimple p G : p.-abelem G -> G :!=: 1 -> charsimple G.

Proof. move=> abelG ntG; apply/charsimpleP; split=> // K ntK /charP[sKG chK]. case/eqVproper: sKG => // /properP[sKG [x Gx notKx]]. have ox := abelem_order_p abelG Gx (group1_contra notKx). have [A [sAG oA defA]] := p_abelem_split1 abelG Gx. case/trivgPn: ntK => y Ky nty; have Gy := subsetP sKG y Ky. have{nty} oy := ab...

Lemma

abelem_charsimple

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "abelem", "abelemS", "abelem_order_p", "apply", "charP", "charsimple", "charsimpleP", "cycle_cyclic", "cycle_subG", "dprodP", "dprodm", "eqVproper", "eqxx", "group1_contra", "injf", "injmSK", "injm_dprodm", "isog", "isogP", "isog_abelem_card", "isog_cyclic_card", "last", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

charsimple_dprod G : charsimple G -> exists H : {group gT}, [/\ H \subset G, simple H & exists2 I : {set {perm gT}}, I \subset Aut G & \big[dprod/1]_(f in I) f @: H = G].

Proof. case/charsimpleP=> ntG simG. have [H minH sHG]: {H : {group gT} | minnormal H G & H \subset G}. by apply: mingroup_exists; rewrite ntG normG. case/mingroupP: minH => /andP[ntH nHG] minH. pose Iok (I : {set {perm gT}}) := (I \subset Aut G) && [exists (M : {group gT} | M <| G), \big[dprod/1]_(f in I) f @: H ...

Lemma

charsimple_dprod

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Aut", "If", "apply", "autm", "autmE", "bigD1", "big_pred1", "bigcupP", "bigcup_max", "bigcupsP", "bigdprodWY", "card_imset", "centS", "cent_joinEr", "centsC", "cents_norm", "characteristic", "charsimple", "charsimpleP", "commG1P", "commg_subl", "commg_subr", "defG", "d...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

simple_sol_prime G : solvable G -> simple G -> prime #|G|.

Proof. move=> solG /simpleP[ntG simG]. have{solG} cGG: abelian G. apply/commG1P; case/simG: (der_normal 1 G) => // /eqP/idPn[]. by rewrite proper_neq // (sol_der1_proper solG). case: (trivgVpdiv G) ntG => [-> | [p p_pr]]; first by rewrite eqxx. case/Cauchy=> // x Gx oxp _; move: p_pr; rewrite -oxp orderE. have: <[x...

Lemma

simple_sol_prime

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "Cauchy", "abelian", "apply", "cGG", "cards1", "commG1P", "cycle_subG", "der_normal", "eqxx", "orderE", "p_pr", "prime", "proper_neq", "simple", "simpleP", "sol_der1_proper", "solvable", "sub_abelian_normal", "trivgVpdiv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

charsimple_solvable G : charsimple G -> solvable G -> is_abelem G.

Proof. case/charsimple_dprod=> H [sHG simH [I Aut_I defG]] solG. have p_pr: prime #|H| by apply: simple_sol_prime (solvableS sHG solG) simH. set p := #|H| in p_pr; apply/is_abelemP; exists p => //. elim/big_rec: _ (G) defG => [_ <-|f B If IH_B M defM]; first exact: abelem1. have [Af [[_ K _ defB] _ _ _]] := (subsetP Au...

Lemma

charsimple_solvable

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "If", "abelem1", "abelemE", "apply", "autmE", "big_rec", "charsimple", "charsimple_dprod", "cyclic_abelian", "defG", "dprodP", "dprod_abelem", "exponent_dvdn", "is_abelem", "is_abelemP", "morphimEsub", "morphim_abelem", "p_pr", "prime", "prime_cyclic", "sHG", "simple_sol_pr...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minnormal_solvable L G H : minnormal H L -> H \subset G -> solvable G -> [/\ L \subset 'N(H), H :!=: 1 & is_abelem H].

Proof. move=> minH sHG solG; have /andP[ntH nHL] := mingroupp minH. split=> //; apply: (charsimple_solvable (minnormal_charsimple minH)). exact: solvableS solG. Qed.

Lemma

minnormal_solvable

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "apply", "charsimple_solvable", "is_abelem", "mingroupp", "minnormal", "minnormal_charsimple", "sHG", "solvable", "solvableS", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

solvable_norm_abelem L G : solvable G -> G <| L -> G :!=: 1 -> exists H : {group gT}, [/\ H \subset G, H <| L, H :!=: 1 & is_abelem H].

Proof. move=> solG /andP[sGL nGL] ntG. have [H minH sHG]: {H : {group gT} | minnormal H L & H \subset G}. by apply: mingroup_exists; rewrite ntG. have [nHL ntH abH] := minnormal_solvable minH sHG solG. by exists H; split; rewrite // /normal (subset_trans sHG). Qed.

Lemma

solvable_norm_abelem

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "apply", "gT", "group", "is_abelem", "mingroup_exists", "minnormal", "minnormal_solvable", "normal", "sHG", "solvable", "split", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivg_Fitting G : solvable G -> ('F(G) == 1) = (G :==: 1).

Proof. move=> solG; apply/idP/idP=> [F1 | /eqP->]; last by rewrite gF1. apply/idPn=> /(solvable_norm_abelem solG (normal_refl _))[M [_ nsMG ntM]]. case/is_abelemP=> p _ /and3P[pM _ _]; case/negP: ntM. by rewrite -subG1 -(eqP F1) Fitting_max ?(pgroup_nil pM). Qed.

Lemma

trivg_Fitting

solvable

solvable/maximal.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...

[ "F1", "Fitting_max", "apply", "gF1", "is_abelemP", "last", "normal_refl", "pgroup_nil", "solvable", "solvable_norm_abelem", "subG1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d