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
(nBA : A \subset 'N(B)) (nCA : A \subset 'N(C)).		Hypotheses	nBA	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norms_gen : A \subset 'N(<<B>>).	Proof. exact: subset_trans nBA (norm_gen B). Qed.	Lemma	norms_gen	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "nBA", "norm_gen", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norms_norm : A \subset 'N('N(B)).	Proof. by apply/normsP=> x Ax; rewrite -normJ (normsP nBA). Qed.	Lemma	norms_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "nBA", "normJ", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsI : A \subset 'N(B :&: C).	Proof. by apply/normsP=> x Ax; rewrite conjIg !(normsP _ x Ax). Qed.	Lemma	normsI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjIg", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsU : A \subset 'N(B :\|: C).	Proof. by apply/normsP=> x Ax; rewrite conjUg !(normsP _ x Ax). Qed.	Lemma	normsU	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjUg", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsIs : B \subset 'N(D) -> A :&: B \subset 'N(C :&: D).	Proof. move/normsP=> nDB; apply/normsP=> x; case/setIP=> Ax Bx. by rewrite conjIg (normsP nCA) ?nDB. Qed.	Lemma	normsIs	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjIg", "normsP", "setIP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsD : A \subset 'N(B :\: C).	Proof. by apply/normsP=> x Ax; rewrite conjDg !(normsP _ x Ax). Qed.	Lemma	normsD	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjDg", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsM : A \subset 'N(B * C).	Proof. by apply/normsP=> x Ax; rewrite conjsMg !(normsP _ x Ax). Qed.	Lemma	normsM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjsMg", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsY : A \subset 'N(B <*> C).	Proof. by apply/normsP=> x Ax; rewrite -genJ conjUg !(normsP _ x Ax). Qed.	Lemma	normsY	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjUg", "genJ", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsR : A \subset 'N([~: B, C]).	Proof. by apply/normsP=> x Ax; rewrite conjsRg !(normsP _ x Ax). Qed.	Lemma	normsR	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjsRg", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norms_class_support : A \subset 'N(class_support B C).	Proof. apply/subsetP=> x Ax; rewrite inE sub_conjg class_supportEr. apply/bigcupsP=> y Cy; rewrite -sub_conjg -conjsgM conjgC conjsgM. by rewrite (normsP nBA) // bigcup_sup ?memJ_norm ?(subsetP nCA). Qed.	Lemma	norms_class_support	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "bigcup_sup", "bigcupsP", "class_support", "class_supportEr", "conjgC", "conjsgM", "inE", "memJ_norm", "nBA", "normsP", "sub_conjg", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsIG A B G : A \subset 'N(B) -> A :&: G \subset 'N(B :&: G).	Proof. by move/normsIs->; rewrite ?normG. Qed.	Lemma	normsIG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normG", "normsIs" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsGI A B G : A \subset 'N(B) -> G :&: A \subset 'N(G :&: B).	Proof. by move=> nBA; rewrite !(setIC G) normsIG. Qed.	Lemma	normsGI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "nBA", "normsIG", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norms_bigcap I r (P : pred I) A (B_ : I -> {set gT}) : A \subset \bigcap_(i <- r \| P i) 'N(B_ i) -> A \subset 'N(\bigcap_(i <- r \| P i) B_ i).	Proof. elim/big_rec2: _ => [\|i B N _ IH /subsetIP[nBiA /IH]]; last exact: normsI. by rewrite normT. Qed.	Lemma	norms_bigcap	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "big_rec2", "gT", "last", "normT", "normsI", "subsetIP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norms_bigcup I r (P : pred I) A (B_ : I -> {set gT}) : A \subset \bigcap_(i <- r \| P i) 'N(B_ i) -> A \subset 'N(\bigcup_(i <- r \| P i) B_ i).	Proof. move=> nBA; rewrite -normCs setC_bigcup norms_bigcap //. by rewrite (eq_bigr _ (fun _ _ => normCs _)). Qed.	Lemma	norms_bigcup	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "eq_bigr", "gT", "nBA", "normCs", "norms_bigcap", "setC_bigcup" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsD1 A B : A \subset 'N(B) -> A \subset 'N(B^#).	Proof. by move/normsD->; rewrite ?norms1. Qed.	Lemma	normsD1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "norms1", "normsD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normD1 A : 'N(A^#) = 'N(A).	Proof. apply/eqP; rewrite eqEsubset normsD1 //. rewrite -{2}(setID A 1) setIC normsU //; apply/normsP=> x _; apply/setP=> y. by rewrite conjIg conjs1g !inE mem_conjg; case: eqP => // ->; rewrite conj1g. Qed.	Lemma	normD1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conj1g", "conjIg", "conjs1g", "eqEsubset", "inE", "mem_conjg", "normsD1", "normsP", "normsU", "setIC", "setID", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalP A B : reflect (A \subset B /\ {in B, normalised A}) (A <\| B).	Proof. by apply: (iffP andP)=> [] [sAB]; move/normsP. Qed.	Lemma	normalP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "normalised", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_sub A B : A <\| B -> A \subset B.	Proof. by case/andP. Qed.	Lemma	normal_sub	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_norm A B : A <\| B -> B \subset 'N(A).	Proof. by case/andP. Qed.	Lemma	normal_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalS G H K : K \subset H -> H \subset G -> K <\| G -> K <\| H.	Proof. by move=> sKH sHG /andP[_ nKG]; rewrite /(K <\| _) sKH (subset_trans sHG). Qed.	Lemma	normalS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "nKG", "sHG", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal1 G : 1 <\| G.	Proof. by rewrite /normal sub1set group1 norms1. Qed.	Lemma	normal1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group1", "normal", "norms1", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_refl G : G <\| G.	Proof. by rewrite /(G <\| _) normG subxx. Qed.	Lemma	normal_refl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normG", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalG G : G <\| 'N(G).	Proof. by rewrite /(G <\| _) normG subxx. Qed.	Lemma	normalG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normG", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalSG G H : H \subset G -> H <\| 'N_G(H).	Proof. by move=> sHG; rewrite /normal subsetI sHG normG subsetIr. Qed.	Lemma	normalSG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normG", "normal", "sHG", "subsetI", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalJ A B x : (A :^ x <\| B :^ x) = (A <\| B).	Proof. by rewrite /normal normJ !conjSg. Qed.	Lemma	normalJ	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjSg", "normJ", "normal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalM G A B : A <\| G -> B <\| G -> A * B <\| G.	Proof. by case/andP=> sAG nAG /andP[sBG nBG]; rewrite /normal mul_subG ?normsM. Qed.	Lemma	normalM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mul_subG", "normal", "normsM", "sAG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalY G A B : A <\| G -> B <\| G -> A <*> B <\| G.	Proof. by case/andP=> sAG ? /andP[sBG ?]; rewrite /normal join_subG sAG sBG ?normsY. Qed.	Lemma	normalY	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "join_subG", "normal", "normsY", "sAG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalYl G H : (H <\| H <*> G) = (G \subset 'N(H)).	Proof. by rewrite /normal joing_subl join_subG normG. Qed.	Lemma	normalYl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "join_subG", "joing_subl", "normG", "normal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalYr G H : (H <\| G <*> H) = (G \subset 'N(H)).	Proof. by rewrite joingC normalYl. Qed.	Lemma	normalYr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joingC", "normalYl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalI G A B : A <\| G -> B <\| G -> A :&: B <\| G.	Proof. by case/andP=> sAG nAG /andP[_ nBG]; rewrite /normal subIset ?sAG // normsI. Qed.	Lemma	normalI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normal", "normsI", "sAG", "subIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_normalI G A : G \subset 'N(A) -> G :&: A <\| G.	Proof. by move=> nAG; rewrite /normal subsetIl normsI ?normG. Qed.	Lemma	norm_normalI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normG", "normal", "normsI", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalGI G H A : H \subset G -> A <\| G -> H :&: A <\| H.	Proof. by move=> sHG /andP[_ nAG]; apply: norm_normalI (subset_trans sHG nAG). Qed.	Lemma	normalGI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "norm_normalI", "sHG", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_subnorm G H : (H <\| 'N_G(H)) = (H \subset G).	Proof. by rewrite /normal subsetIr subsetI normG !andbT. Qed.	Lemma	normal_subnorm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normG", "normal", "subsetI", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalD1 A G : (A^# <\| G) = (A <\| G).	Proof. by rewrite /normal normD1 subDset (setUidPr (sub1G G)). Qed.	Lemma	normalD1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normD1", "normal", "setUidPr", "sub1G", "subDset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcore_sub A G : gcore A G \subset A.	Proof. by rewrite (bigcap_min 1) ?conjsg1. Qed.	Lemma	gcore_sub	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "bigcap_min", "conjsg1", "gcore" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcore_norm A G : G \subset 'N(gcore A G).	Proof. apply/subsetP=> x Gx; rewrite inE; apply/bigcapsP=> y Gy. by rewrite sub_conjg -conjsgM bigcap_inf ?groupM ?groupV. Qed.	Lemma	gcore_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "bigcap_inf", "bigcapsP", "conjsgM", "gcore", "groupM", "groupV", "inE", "sub_conjg", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcore_normal A G : A \subset G -> gcore A G <\| G.	Proof. by move=> sAG; rewrite /normal gcore_norm (subset_trans (gcore_sub A G)). Qed.	Lemma	gcore_normal	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gcore", "gcore_norm", "gcore_sub", "normal", "sAG", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcore_max A B G : B \subset A -> G \subset 'N(B) -> B \subset gcore A G.	Proof. move=> sBA nBG; apply/bigcapsP=> y Gy. by rewrite -sub_conjgV (normsP nBG) ?groupV. Qed.	Lemma	gcore_max	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "bigcapsP", "gcore", "groupV", "normsP", "sub_conjgV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_gcore A B G : G \subset 'N(B) -> (B \subset gcore A G) = (B \subset A).	Proof. move=> nBG; apply/idP/idP=> [sBAG \| sBA]; last exact: gcore_max. exact: subset_trans (gcore_sub A G). Qed.	Lemma	sub_gcore	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "gcore", "gcore_max", "gcore_sub", "last", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_index2 G H x : H \subset G -> #\|G : H\| = 2 -> x \in G :\: H -> H :* x = G :\: H.	Proof. move=> sHG indexHG => /setDP[Gx notHx]; apply/eqP. rewrite eqEcard -(leq_add2l #\|G :&: H\|) cardsID -(LagrangeI G H) indexHG muln2. rewrite (setIidPr sHG) card_rcoset addnn leqnn andbT. apply/subsetP=> _ /rcosetP[y Hy ->]; apply/setDP. by rewrite !groupMl // (subsetP sHG). Qed.	Lemma	rcoset_index2	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "LagrangeI", "addnn", "apply", "card_rcoset", "cardsID", "eqEcard", "groupMl", "leq_add2l", "leqnn", "muln2", "rcosetP", "sHG", "setDP", "setIidPr", "subsetP" ]	the coset is equal to the complement is used in extremal.v.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index2_normal G H : H \subset G -> #\|G : H\| = 2 -> H <\| G.	Proof. move=> sHG indexHG; rewrite /normal sHG; apply/subsetP=> x Gx. case Hx: (x \in H); first by rewrite inE conjGid. rewrite inE conjsgE mulgA -sub_rcosetV -invg_rcoset. by rewrite !(rcoset_index2 sHG) ?inE ?groupV ?Hx // invDg !invGid. Qed.	Lemma	index2_normal	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjGid", "conjsgE", "groupV", "inE", "invDg", "invGid", "invg_rcoset", "mulgA", "normal", "rcoset_index2", "sHG", "sub_rcosetV", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1P x y : reflect (commute x y) (x \in 'C[y]).	Proof. rewrite [x \in _]inE conjg_set1 sub1set !inE (sameP eqP conjg_fixP)commg1_sym. exact: commgP. Qed.	Lemma	cent1P	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "commg1_sym", "commgP", "commute", "conjg_fixP", "conjg_set1", "inE", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1id x : x \in 'C[x].	Proof. exact/cent1P. Qed.	Lemma	cent1id	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cent1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1E x y : (x \in 'C[y]) = (x * y == y * x).	Proof. by rewrite (sameP (cent1P x y) eqP). Qed.	Lemma	cent1E	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cent1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1C x y : (x \in 'C[y]) = (y \in 'C[x]).	Proof. by rewrite !cent1E eq_sym. Qed.	Lemma	cent1C	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cent1E", "eq_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centraliser_group A : {group _}	:= Eval hnf in [group of 'C(A)].	Canonical	centraliser_group	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_set1 x : 'C([set x]) = 'C[x].	Proof. by apply: big_pred1 => y /=; rewrite !inE. Qed.	Lemma	cent_set1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "big_pred1", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1J x y : 'C[x ^ y] = 'C[x] :^ y.	Proof. by rewrite -conjg_set1 normJ. Qed.	Lemma	cent1J	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjg_set1", "normJ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centP A x : reflect (centralises x A) (x \in 'C(A)).	Proof. by apply: (iffP bigcapP) => cxA y /cxA/cent1P. Qed.	Lemma	centP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "bigcapP", "cent1P", "centralises" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centsP A B : reflect {in A, centralised B} (A \subset 'C(B)).	Proof. by apply: (iffP subsetP) => cAB x /cAB/centP. Qed.	Lemma	centsP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centP", "centralised", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centsC A B : (A \subset 'C(B)) = (B \subset 'C(A)).	Proof. by apply/centsP/centsP=> cAB x ? y ?; rewrite /commute -cAB. Qed.	Lemma	centsC	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centsP", "commute" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cents1 A : A \subset 'C(1).	Proof. by rewrite centsC sub1G. Qed.	Lemma	cents1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "centsC", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1T : 'C(1) = setT :> {set gT}.	Proof. by apply/eqP; rewrite -subTset cents1. Qed.	Lemma	cent1T	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cents1", "gT", "setT", "subTset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent11T : 'C[1] = setT :> {set gT}.	Proof. by rewrite -cent_set1 cent1T. Qed.	Lemma	cent11T	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cent1T", "cent_set1", "gT", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_sub A : 'C(A) \subset 'N(A).	Proof. apply/subsetP=> x /centP cAx; rewrite inE. by apply/subsetP=> _ /imsetP[y Ay ->]; rewrite /conjg -cAx ?mulKg. Qed.	Lemma	cent_sub	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centP", "conjg", "imsetP", "inE", "mulKg", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cents_norm A B : A \subset 'C(B) -> A \subset 'N(B).	Proof. by move=> cAB; apply: subset_trans (cent_sub B). Qed.	Lemma	cents_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cent_sub", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centC A B : A \subset 'C(B) -> commute A B.	Proof. by move=> cAB; apply: normC (cents_norm cAB). Qed.	Lemma	centC	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cents_norm", "commute", "normC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_joinEl G H : G \subset 'C(H) -> G <> H = G H.	Proof. by move=> cGH; apply: norm_joinEl (cents_norm cGH). Qed.	Lemma	cent_joinEl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cents_norm", "norm_joinEl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_joinEr G H : H \subset 'C(G) -> G <> H = G H.	Proof. by move=> cGH; apply: norm_joinEr (cents_norm cGH). Qed.	Lemma	cent_joinEr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cents_norm", "norm_joinEr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centJ A x : 'C(A :^ x) = 'C(A) :^ x.	Proof. apply/setP=> y; rewrite mem_conjg; apply/centP/centP=> cAy z Az. apply: (conjg_inj x). by rewrite conjMg [in RHS]conjMg conjgKV cAy ?memJ_conjg. by apply: (conjg_inj x^-1); rewrite 2!conjMg cAy -?mem_conjg. Qed.	Lemma	centJ	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centP", "conjMg", "conjgKV", "conjg_inj", "memJ_conjg", "mem_conjg", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_norm A : 'N(A) \subset 'N('C(A)).	Proof. by apply/normsP=> x nCx; rewrite -centJ (normP nCx). Qed.	Lemma	cent_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centJ", "normP", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norms_cent A B : A \subset 'N(B) -> A \subset 'N('C(B)).	Proof. by move=> nBA; apply: subset_trans nBA (cent_norm B). Qed.	Lemma	norms_cent	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cent_norm", "nBA", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_normal A : 'C(A) <\| 'N(A).	Proof. by rewrite /(_ <\| _) cent_sub cent_norm. Qed.	Lemma	cent_normal	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cent_norm", "cent_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centS A B : B \subset A -> 'C(A) \subset 'C(B).	Proof. by move=> sAB; rewrite centsC (subset_trans sAB) 1?centsC. Qed.	Lemma	centS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "centsC", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centsS A B C : A \subset B -> C \subset 'C(B) -> C \subset 'C(A).	Proof. by move=> sAB cCB; apply: subset_trans cCB (centS sAB). Qed.	Lemma	centsS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centS", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centSS A B C D : A \subset C -> B \subset D -> C \subset 'C(D) -> A \subset 'C(B).	Proof. by move=> sAC sBD cCD; apply: subset_trans (centsS sBD cCD). Qed.	Lemma	centSS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centsS", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centI A B : 'C(A) <*> 'C(B) \subset 'C(A :&: B).	Proof. by rewrite gen_subG subUset !centS ?(subsetIl, subsetIr). Qed.	Lemma	centI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "centS", "gen_subG", "subUset", "subsetIl", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centU A B : 'C(A :\|: B) = 'C(A) :&: 'C(B).	Proof. apply/eqP; rewrite eqEsubset subsetI 2?centS ?(subsetUl, subsetUr) //=. by rewrite centsC subUset -centsC subsetIl -centsC subsetIr. Qed.	Lemma	centU	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centS", "centsC", "eqEsubset", "subUset", "subsetI", "subsetIl", "subsetIr", "subsetUl", "subsetUr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_gen A : 'C(<<A>>) = 'C(A).	Proof. by apply/setP=> x; rewrite -!sub1set centsC gen_subG centsC. Qed.	Lemma	cent_gen	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centsC", "gen_subG", "setP", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_cycle x : 'C(<[x]>) = 'C[x].	Proof. by rewrite cent_gen cent_set1. Qed.	Lemma	cent_cycle	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cent_gen", "cent_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_cent1 A x : (A \subset 'C[x]) = (x \in 'C(A)).	Proof. by rewrite -cent_cycle centsC cycle_subG. Qed.	Lemma	sub_cent1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cent_cycle", "centsC", "cycle_subG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cents_cycle x y : commute x y -> <[x]> \subset 'C(<[y]>).	Proof. by move=> cxy; rewrite cent_cycle cycle_subG; apply/cent1P. Qed.	Lemma	cents_cycle	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cent1P", "cent_cycle", "commute", "cycle_subG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle_abelian x : abelian <[x]>.	Proof. exact: cents_cycle. Qed.	Lemma	cycle_abelian	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "abelian", "cents_cycle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centY A B : 'C(A <*> B) = 'C(A) :&: 'C(B).	Proof. by rewrite cent_gen centU. Qed.	Lemma	centY	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "centU", "cent_gen" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centM G H : 'C(G * H) = 'C(G) :&: 'C(H).	Proof. by rewrite -cent_gen genM_join centY. Qed.	Lemma	centM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "centY", "cent_gen", "genM_join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_classP x G : reflect (x ^: G = [set x]) (x \in 'C(G)).	Proof. apply: (iffP (centP _ _)) => [Cx \| Cx1 y Gy]. apply/eqP; rewrite eqEsubset sub1set class_refl andbT. by apply/subsetP=> _ /imsetP[y Gy ->]; rewrite !inE conjgE Cx ?mulKg. by apply/commgP/conjg_fixP/set1P; rewrite -Cx1; apply/imsetP; exists y. Qed.	Lemma	cent_classP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centP", "class_refl", "commgP", "conjgE", "conjg_fixP", "eqEsubset", "imsetP", "inE", "mulKg", "set1P", "sub1set", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commG1P A B : reflect ([~: A, B] = 1) (A \subset 'C(B)).	Proof. apply: (iffP (centsP A B)) => [cAB \| cAB1 x Ax y By]. apply/trivgP; rewrite gen_subG; apply/subsetP=> _ /imset2P[x y Ax Ay ->]. by rewrite inE; apply/commgP; apply: cAB. by apply/commgP; rewrite -in_set1 -[[set 1]]cAB1 mem_commg. Qed.	Lemma	commG1P	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "centsP", "commgP", "gen_subG", "imset2P", "inE", "in_set1", "mem_commg", "subsetP", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelianE A : abelian A = (A \subset 'C(A)).	Proof. by []. Qed.	Lemma	abelianE	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "abelian" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian1 : abelian [1 gT].	Proof. exact: sub1G. Qed.	Lemma	abelian1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "abelian", "gT", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelianS A B : A \subset B -> abelian B -> abelian A.	Proof. by move=> sAB; apply: centSS. Qed.	Lemma	abelianS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "abelian", "apply", "centSS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelianJ A x : abelian (A :^ x) = abelian A.	Proof. by rewrite /abelian centJ conjSg. Qed.	Lemma	abelianJ	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "abelian", "centJ", "conjSg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_gen A : abelian <<A>> = abelian A.	Proof. by rewrite /abelian cent_gen gen_subG. Qed.	Lemma	abelian_gen	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "abelian", "cent_gen", "gen_subG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelianY A B : abelian (A <*> B) = [&& abelian A, abelian B & B \subset 'C(A)].	Proof. rewrite /abelian join_subG /= centY !subsetI -!andbA; congr (_ && _). by rewrite centsC andbA andbb andbC. Qed.	Lemma	abelianY	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "abelian", "centY", "centsC", "join_subG", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelianM G H : abelian (G * H) = [&& abelian G, abelian H & H \subset 'C(G)].	Proof. by rewrite -abelian_gen genM_join abelianY. Qed.	Lemma	abelianM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "abelian", "abelianY", "abelian_gen", "genM_join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cAA : abelian A.		Hypothesis	cAA	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "abelian" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_abelian_cent : C \subset A -> A \subset 'C(C).	Proof. by move=> sCA; rewrite centsC (subset_trans sCA). Qed.	Lemma	sub_abelian_cent	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "centsC", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_abelian_cent2 : B \subset A -> C \subset A -> B \subset 'C(C).	Proof. by move=> sBA; move/sub_abelian_cent; apply: subset_trans. Qed.	Lemma	sub_abelian_cent2	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "sub_abelian_cent", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_abelian_norm : C \subset A -> A \subset 'N(C).	Proof. by move=> sCA; rewrite cents_norm ?sub_abelian_cent. Qed.	Lemma	sub_abelian_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cents_norm", "sub_abelian_cent" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_abelian_normal : (C \subset A) = (C <\| A).	Proof. by rewrite /normal; case sHG: (C \subset A); rewrite // sub_abelian_norm. Qed.	Lemma	sub_abelian_normal	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normal", "sHG", "sub_abelian_norm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''N' ( A )"	:= (normaliser_group A) : Group_scope.	Notation	''N' ( A )	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normaliser_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C' ( A )"	:= (centraliser_group A) : Group_scope.	Notation	''C' ( A )	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "centraliser_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C' [ x ]"	:= (normaliser_group [set x%g]) : Group_scope.	Notation	''C' [ x ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normaliser_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''N_' G ( A )"	:= (setI_group G 'N(A)) : Group_scope.	Notation	''N_' G ( A )	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "setI_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' G ( A )"	:= (setI_group G 'C(A)) : Group_scope.	Notation	''C_' G ( A )	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "setI_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( G ) ( A )"	:= (setI_group G 'C(A)) (only parsing) : Group_scope.	Notation	''C_' ( G ) ( A )	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "setI_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' G [ x ]"	:= (setI_group G 'C[x]) : Group_scope.	Notation	''C_' G [ x ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "setI_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( G ) [ x ]"	:= (setI_group G 'C[x]) (only parsing) : Group_scope.	Notation	''C_' ( G ) [ x ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "setI_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxgroup A gP	:= maxset (fun A => group_set A && gP <<A>>%G) A.	Definition	maxgroup	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group_set", "maxset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mingroup A gP	:= minset (fun A => group_set A && gP <<A>>%G) A.	Definition	mingroup	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group_set", "minset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

(nBA : A \subset 'N(B)) (nCA : A \subset 'N(C)).

Hypotheses

nBA

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norms_gen : A \subset 'N(<<B>>).

Proof. exact: subset_trans nBA (norm_gen B). Qed.

Lemma

norms_gen

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "nBA", "norm_gen", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norms_norm : A \subset 'N('N(B)).

Proof. by apply/normsP=> x Ax; rewrite -normJ (normsP nBA). Qed.

Lemma

norms_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "nBA", "normJ", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsI : A \subset 'N(B :&: C).

Proof. by apply/normsP=> x Ax; rewrite conjIg !(normsP _ x Ax). Qed.

Lemma

normsI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjIg", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsU : A \subset 'N(B :|: C).

Proof. by apply/normsP=> x Ax; rewrite conjUg !(normsP _ x Ax). Qed.

Lemma

normsU

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjUg", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsIs : B \subset 'N(D) -> A :&: B \subset 'N(C :&: D).

Proof. move/normsP=> nDB; apply/normsP=> x; case/setIP=> Ax Bx. by rewrite conjIg (normsP nCA) ?nDB. Qed.

Lemma

normsIs

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjIg", "normsP", "setIP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsD : A \subset 'N(B :\: C).

Proof. by apply/normsP=> x Ax; rewrite conjDg !(normsP _ x Ax). Qed.

Lemma

normsD

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjDg", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsM : A \subset 'N(B * C).

Proof. by apply/normsP=> x Ax; rewrite conjsMg !(normsP _ x Ax). Qed.

Lemma

normsM

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjsMg", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsY : A \subset 'N(B <*> C).

Proof. by apply/normsP=> x Ax; rewrite -genJ conjUg !(normsP _ x Ax). Qed.

Lemma

normsY

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjUg", "genJ", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsR : A \subset 'N([~: B, C]).

Proof. by apply/normsP=> x Ax; rewrite conjsRg !(normsP _ x Ax). Qed.

Lemma

normsR

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjsRg", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norms_class_support : A \subset 'N(class_support B C).

Proof. apply/subsetP=> x Ax; rewrite inE sub_conjg class_supportEr. apply/bigcupsP=> y Cy; rewrite -sub_conjg -conjsgM conjgC conjsgM. by rewrite (normsP nBA) // bigcup_sup ?memJ_norm ?(subsetP nCA). Qed.

Lemma

norms_class_support

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "bigcup_sup", "bigcupsP", "class_support", "class_supportEr", "conjgC", "conjsgM", "inE", "memJ_norm", "nBA", "normsP", "sub_conjg", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsIG A B G : A \subset 'N(B) -> A :&: G \subset 'N(B :&: G).

Proof. by move/normsIs->; rewrite ?normG. Qed.

Lemma

normsIG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normG", "normsIs" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsGI A B G : A \subset 'N(B) -> G :&: A \subset 'N(G :&: B).

Proof. by move=> nBA; rewrite !(setIC G) normsIG. Qed.

Lemma

normsGI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "nBA", "normsIG", "setIC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norms_bigcap I r (P : pred I) A (B_ : I -> {set gT}) : A \subset \bigcap_(i <- r | P i) 'N(B_ i) -> A \subset 'N(\bigcap_(i <- r | P i) B_ i).

Proof. elim/big_rec2: _ => [|i B N _ IH /subsetIP[nBiA /IH]]; last exact: normsI. by rewrite normT. Qed.

Lemma

norms_bigcap

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "big_rec2", "gT", "last", "normT", "normsI", "subsetIP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norms_bigcup I r (P : pred I) A (B_ : I -> {set gT}) : A \subset \bigcap_(i <- r | P i) 'N(B_ i) -> A \subset 'N(\bigcup_(i <- r | P i) B_ i).

Proof. move=> nBA; rewrite -normCs setC_bigcup norms_bigcap //. by rewrite (eq_bigr _ (fun _ _ => normCs _)). Qed.

Lemma

norms_bigcup

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "eq_bigr", "gT", "nBA", "normCs", "norms_bigcap", "setC_bigcup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsD1 A B : A \subset 'N(B) -> A \subset 'N(B^#).

Proof. by move/normsD->; rewrite ?norms1. Qed.

Lemma

normsD1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "norms1", "normsD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normD1 A : 'N(A^#) = 'N(A).

Proof. apply/eqP; rewrite eqEsubset normsD1 //. rewrite -{2}(setID A 1) setIC normsU //; apply/normsP=> x _; apply/setP=> y. by rewrite conjIg conjs1g !inE mem_conjg; case: eqP => // ->; rewrite conj1g. Qed.

Lemma

normD1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conj1g", "conjIg", "conjs1g", "eqEsubset", "inE", "mem_conjg", "normsD1", "normsP", "normsU", "setIC", "setID", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalP A B : reflect (A \subset B /\ {in B, normalised A}) (A <| B).

Proof. by apply: (iffP andP)=> [] [sAB]; move/normsP. Qed.

Lemma

normalP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "normalised", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal_sub A B : A <| B -> A \subset B.

Proof. by case/andP. Qed.

Lemma

normal_sub

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal_norm A B : A <| B -> B \subset 'N(A).

Proof. by case/andP. Qed.

Lemma

normal_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalS G H K : K \subset H -> H \subset G -> K <| G -> K <| H.

Proof. by move=> sKH sHG /andP[_ nKG]; rewrite /(K <| _) sKH (subset_trans sHG). Qed.

Lemma

normalS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "nKG", "sHG", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal1 G : 1 <| G.

Proof. by rewrite /normal sub1set group1 norms1. Qed.

Lemma

normal1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group1", "normal", "norms1", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal_refl G : G <| G.

Proof. by rewrite /(G <| _) normG subxx. Qed.

Lemma

normal_refl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normG", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalG G : G <| 'N(G).

Proof. by rewrite /(G <| _) normG subxx. Qed.

Lemma

normalG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normG", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalSG G H : H \subset G -> H <| 'N_G(H).

Proof. by move=> sHG; rewrite /normal subsetI sHG normG subsetIr. Qed.

Lemma

normalSG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normG", "normal", "sHG", "subsetI", "subsetIr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalJ A B x : (A :^ x <| B :^ x) = (A <| B).

Proof. by rewrite /normal normJ !conjSg. Qed.

Lemma

normalJ

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "conjSg", "normJ", "normal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalM G A B : A <| G -> B <| G -> A * B <| G.

Proof. by case/andP=> sAG nAG /andP[sBG nBG]; rewrite /normal mul_subG ?normsM. Qed.

Lemma

normalM

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "mul_subG", "normal", "normsM", "sAG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalY G A B : A <| G -> B <| G -> A <*> B <| G.

Proof. by case/andP=> sAG ? /andP[sBG ?]; rewrite /normal join_subG sAG sBG ?normsY. Qed.

Lemma

normalY

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "join_subG", "normal", "normsY", "sAG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalYl G H : (H <| H <*> G) = (G \subset 'N(H)).

Proof. by rewrite /normal joing_subl join_subG normG. Qed.

Lemma

normalYl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "join_subG", "joing_subl", "normG", "normal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalYr G H : (H <| G <*> H) = (G \subset 'N(H)).

Proof. by rewrite joingC normalYl. Qed.

Lemma

normalYr

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joingC", "normalYl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalI G A B : A <| G -> B <| G -> A :&: B <| G.

Proof. by case/andP=> sAG nAG /andP[_ nBG]; rewrite /normal subIset ?sAG // normsI. Qed.

Lemma

normalI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normal", "normsI", "sAG", "subIset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_normalI G A : G \subset 'N(A) -> G :&: A <| G.

Proof. by move=> nAG; rewrite /normal subsetIl normsI ?normG. Qed.

Lemma

norm_normalI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normG", "normal", "normsI", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalGI G H A : H \subset G -> A <| G -> H :&: A <| H.

Proof. by move=> sHG /andP[_ nAG]; apply: norm_normalI (subset_trans sHG nAG). Qed.

Lemma

normalGI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "norm_normalI", "sHG", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal_subnorm G H : (H <| 'N_G(H)) = (H \subset G).

Proof. by rewrite /normal subsetIr subsetI normG !andbT. Qed.

Lemma

normal_subnorm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normG", "normal", "subsetI", "subsetIr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalD1 A G : (A^# <| G) = (A <| G).

Proof. by rewrite /normal normD1 subDset (setUidPr (sub1G G)). Qed.

Lemma

normalD1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normD1", "normal", "setUidPr", "sub1G", "subDset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gcore_sub A G : gcore A G \subset A.

Proof. by rewrite (bigcap_min 1) ?conjsg1. Qed.

Lemma

gcore_sub

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "bigcap_min", "conjsg1", "gcore" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gcore_norm A G : G \subset 'N(gcore A G).

Proof. apply/subsetP=> x Gx; rewrite inE; apply/bigcapsP=> y Gy. by rewrite sub_conjg -conjsgM bigcap_inf ?groupM ?groupV. Qed.

Lemma

gcore_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "bigcap_inf", "bigcapsP", "conjsgM", "gcore", "groupM", "groupV", "inE", "sub_conjg", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gcore_normal A G : A \subset G -> gcore A G <| G.

Proof. by move=> sAG; rewrite /normal gcore_norm (subset_trans (gcore_sub A G)). Qed.

Lemma

gcore_normal

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gcore", "gcore_norm", "gcore_sub", "normal", "sAG", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gcore_max A B G : B \subset A -> G \subset 'N(B) -> B \subset gcore A G.

Proof. move=> sBA nBG; apply/bigcapsP=> y Gy. by rewrite -sub_conjgV (normsP nBG) ?groupV. Qed.

Lemma

gcore_max

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "bigcapsP", "gcore", "groupV", "normsP", "sub_conjgV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_gcore A B G : G \subset 'N(B) -> (B \subset gcore A G) = (B \subset A).

Proof. move=> nBG; apply/idP/idP=> [sBAG | sBA]; last exact: gcore_max. exact: subset_trans (gcore_sub A G). Qed.

Lemma

sub_gcore

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "gcore", "gcore_max", "gcore_sub", "last", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_index2 G H x : H \subset G -> #|G : H| = 2 -> x \in G :\: H -> H :* x = G :\: H.

Proof. move=> sHG indexHG => /setDP[Gx notHx]; apply/eqP. rewrite eqEcard -(leq_add2l #|G :&: H|) cardsID -(LagrangeI G H) indexHG muln2. rewrite (setIidPr sHG) card_rcoset addnn leqnn andbT. apply/subsetP=> _ /rcosetP[y Hy ->]; apply/setDP. by rewrite !groupMl // (subsetP sHG). Qed.

Lemma

rcoset_index2

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "LagrangeI", "addnn", "apply", "card_rcoset", "cardsID", "eqEcard", "groupMl", "leq_add2l", "leqnn", "muln2", "rcosetP", "sHG", "setDP", "setIidPr", "subsetP" ]

the coset is equal to the complement is used in extremal.v.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

index2_normal G H : H \subset G -> #|G : H| = 2 -> H <| G.

Proof. move=> sHG indexHG; rewrite /normal sHG; apply/subsetP=> x Gx. case Hx: (x \in H); first by rewrite inE conjGid. rewrite inE conjsgE mulgA -sub_rcosetV -invg_rcoset. by rewrite !(rcoset_index2 sHG) ?inE ?groupV ?Hx // invDg !invGid. Qed.

Lemma

index2_normal

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjGid", "conjsgE", "groupV", "inE", "invDg", "invGid", "invg_rcoset", "mulgA", "normal", "rcoset_index2", "sHG", "sub_rcosetV", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1P x y : reflect (commute x y) (x \in 'C[y]).

Proof. rewrite [x \in _]inE conjg_set1 sub1set !inE (sameP eqP conjg_fixP)commg1_sym. exact: commgP. Qed.

Lemma

cent1P

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "commg1_sym", "commgP", "commute", "conjg_fixP", "conjg_set1", "inE", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1id x : x \in 'C[x].

Proof. exact/cent1P. Qed.

Lemma

cent1id

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cent1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1E x y : (x \in 'C[y]) = (x * y == y * x).

Proof. by rewrite (sameP (cent1P x y) eqP). Qed.

Lemma

cent1E

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cent1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1C x y : (x \in 'C[y]) = (y \in 'C[x]).

Proof. by rewrite !cent1E eq_sym. Qed.

Lemma

cent1C

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cent1E", "eq_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centraliser_group A : {group _}

:= Eval hnf in [group of 'C(A)].

Canonical

centraliser_group

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_set1 x : 'C([set x]) = 'C[x].

Proof. by apply: big_pred1 => y /=; rewrite !inE. Qed.

Lemma

cent_set1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "big_pred1", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1J x y : 'C[x ^ y] = 'C[x] :^ y.

Proof. by rewrite -conjg_set1 normJ. Qed.

Lemma

cent1J

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "conjg_set1", "normJ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centP A x : reflect (centralises x A) (x \in 'C(A)).

Proof. by apply: (iffP bigcapP) => cxA y /cxA/cent1P. Qed.

Lemma

centP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "bigcapP", "cent1P", "centralises" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centsP A B : reflect {in A, centralised B} (A \subset 'C(B)).

Proof. by apply: (iffP subsetP) => cAB x /cAB/centP. Qed.

Lemma

centsP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centP", "centralised", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centsC A B : (A \subset 'C(B)) = (B \subset 'C(A)).

Proof. by apply/centsP/centsP=> cAB x ? y ?; rewrite /commute -cAB. Qed.

Lemma

centsC

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centsP", "commute" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cents1 A : A \subset 'C(1).

Proof. by rewrite centsC sub1G. Qed.

Lemma

cents1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "centsC", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1T : 'C(1) = setT :> {set gT}.

Proof. by apply/eqP; rewrite -subTset cents1. Qed.

Lemma

cent1T

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cents1", "gT", "setT", "subTset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent11T : 'C[1] = setT :> {set gT}.

Proof. by rewrite -cent_set1 cent1T. Qed.

Lemma

cent11T

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cent1T", "cent_set1", "gT", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_sub A : 'C(A) \subset 'N(A).

Proof. apply/subsetP=> x /centP cAx; rewrite inE. by apply/subsetP=> _ /imsetP[y Ay ->]; rewrite /conjg -cAx ?mulKg. Qed.

Lemma

cent_sub

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centP", "conjg", "imsetP", "inE", "mulKg", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cents_norm A B : A \subset 'C(B) -> A \subset 'N(B).

Proof. by move=> cAB; apply: subset_trans (cent_sub B). Qed.

Lemma

cents_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cent_sub", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centC A B : A \subset 'C(B) -> commute A B.

Proof. by move=> cAB; apply: normC (cents_norm cAB). Qed.

Lemma

centC

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cents_norm", "commute", "normC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_joinEl G H : G \subset 'C(H) -> G <*> H = G * H.

Proof. by move=> cGH; apply: norm_joinEl (cents_norm cGH). Qed.

Lemma

cent_joinEl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cents_norm", "norm_joinEl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_joinEr G H : H \subset 'C(G) -> G <*> H = G * H.

Proof. by move=> cGH; apply: norm_joinEr (cents_norm cGH). Qed.

Lemma

cent_joinEr

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cents_norm", "norm_joinEr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centJ A x : 'C(A :^ x) = 'C(A) :^ x.

Proof. apply/setP=> y; rewrite mem_conjg; apply/centP/centP=> cAy z Az. apply: (conjg_inj x). by rewrite conjMg [in RHS]conjMg conjgKV cAy ?memJ_conjg. by apply: (conjg_inj x^-1); rewrite 2!conjMg cAy -?mem_conjg. Qed.

Lemma

centJ

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centP", "conjMg", "conjgKV", "conjg_inj", "memJ_conjg", "mem_conjg", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_norm A : 'N(A) \subset 'N('C(A)).

Proof. by apply/normsP=> x nCx; rewrite -centJ (normP nCx). Qed.

Lemma

cent_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centJ", "normP", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norms_cent A B : A \subset 'N(B) -> A \subset 'N('C(B)).

Proof. by move=> nBA; apply: subset_trans nBA (cent_norm B). Qed.

Lemma

norms_cent

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cent_norm", "nBA", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_normal A : 'C(A) <| 'N(A).

Proof. by rewrite /(_ <| _) cent_sub cent_norm. Qed.

Lemma

cent_normal

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cent_norm", "cent_sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centS A B : B \subset A -> 'C(A) \subset 'C(B).

Proof. by move=> sAB; rewrite centsC (subset_trans sAB) 1?centsC. Qed.

Lemma

centS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "centsC", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centsS A B C : A \subset B -> C \subset 'C(B) -> C \subset 'C(A).

Proof. by move=> sAB cCB; apply: subset_trans cCB (centS sAB). Qed.

Lemma

centsS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centS", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centSS A B C D : A \subset C -> B \subset D -> C \subset 'C(D) -> A \subset 'C(B).

Proof. by move=> sAC sBD cCD; apply: subset_trans (centsS sBD cCD). Qed.

Lemma

centSS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centsS", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centI A B : 'C(A) <*> 'C(B) \subset 'C(A :&: B).

Proof. by rewrite gen_subG subUset !centS ?(subsetIl, subsetIr). Qed.

Lemma

centI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "centS", "gen_subG", "subUset", "subsetIl", "subsetIr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centU A B : 'C(A :|: B) = 'C(A) :&: 'C(B).

Proof. apply/eqP; rewrite eqEsubset subsetI 2?centS ?(subsetUl, subsetUr) //=. by rewrite centsC subUset -centsC subsetIl -centsC subsetIr. Qed.

Lemma

centU

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centS", "centsC", "eqEsubset", "subUset", "subsetI", "subsetIl", "subsetIr", "subsetUl", "subsetUr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_gen A : 'C(<<A>>) = 'C(A).

Proof. by apply/setP=> x; rewrite -!sub1set centsC gen_subG centsC. Qed.

Lemma

cent_gen

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centsC", "gen_subG", "setP", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_cycle x : 'C(<[x]>) = 'C[x].

Proof. by rewrite cent_gen cent_set1. Qed.

Lemma

cent_cycle

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cent_gen", "cent_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_cent1 A x : (A \subset 'C[x]) = (x \in 'C(A)).

Proof. by rewrite -cent_cycle centsC cycle_subG. Qed.

Lemma

sub_cent1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cent_cycle", "centsC", "cycle_subG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cents_cycle x y : commute x y -> <[x]> \subset 'C(<[y]>).

Proof. by move=> cxy; rewrite cent_cycle cycle_subG; apply/cent1P. Qed.

Lemma

cents_cycle

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cent1P", "cent_cycle", "commute", "cycle_subG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle_abelian x : abelian <[x]>.

Proof. exact: cents_cycle. Qed.

Lemma

cycle_abelian

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "abelian", "cents_cycle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centY A B : 'C(A <*> B) = 'C(A) :&: 'C(B).

Proof. by rewrite cent_gen centU. Qed.

Lemma

centY

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "centU", "cent_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centM G H : 'C(G * H) = 'C(G) :&: 'C(H).

Proof. by rewrite -cent_gen genM_join centY. Qed.

Lemma

centM

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "centY", "cent_gen", "genM_join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_classP x G : reflect (x ^: G = [set x]) (x \in 'C(G)).

Proof. apply: (iffP (centP _ _)) => [Cx | Cx1 y Gy]. apply/eqP; rewrite eqEsubset sub1set class_refl andbT. by apply/subsetP=> _ /imsetP[y Gy ->]; rewrite !inE conjgE Cx ?mulKg. by apply/commgP/conjg_fixP/set1P; rewrite -Cx1; apply/imsetP; exists y. Qed.

Lemma

cent_classP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centP", "class_refl", "commgP", "conjgE", "conjg_fixP", "eqEsubset", "imsetP", "inE", "mulKg", "set1P", "sub1set", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commG1P A B : reflect ([~: A, B] = 1) (A \subset 'C(B)).

Proof. apply: (iffP (centsP A B)) => [cAB | cAB1 x Ax y By]. apply/trivgP; rewrite gen_subG; apply/subsetP=> _ /imset2P[x y Ax Ay ->]. by rewrite inE; apply/commgP; apply: cAB. by apply/commgP; rewrite -in_set1 -[[set 1]]cAB1 mem_commg. Qed.

Lemma

commG1P

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "centsP", "commgP", "gen_subG", "imset2P", "inE", "in_set1", "mem_commg", "subsetP", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelianE A : abelian A = (A \subset 'C(A)).

Proof. by []. Qed.

Lemma

abelianE

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "abelian" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian1 : abelian [1 gT].

Proof. exact: sub1G. Qed.

Lemma

abelian1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "abelian", "gT", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelianS A B : A \subset B -> abelian B -> abelian A.

Proof. by move=> sAB; apply: centSS. Qed.

Lemma

abelianS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "abelian", "apply", "centSS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelianJ A x : abelian (A :^ x) = abelian A.

Proof. by rewrite /abelian centJ conjSg. Qed.

Lemma

abelianJ

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "abelian", "centJ", "conjSg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_gen A : abelian <<A>> = abelian A.

Proof. by rewrite /abelian cent_gen gen_subG. Qed.

Lemma

abelian_gen

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "abelian", "cent_gen", "gen_subG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelianY A B : abelian (A <*> B) = [&& abelian A, abelian B & B \subset 'C(A)].

Proof. rewrite /abelian join_subG /= centY !subsetI -!andbA; congr (_ && _). by rewrite centsC andbA andbb andbC. Qed.

Lemma

abelianY

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "abelian", "centY", "centsC", "join_subG", "subsetI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelianM G H : abelian (G * H) = [&& abelian G, abelian H & H \subset 'C(G)].

Proof. by rewrite -abelian_gen genM_join abelianY. Qed.

Lemma

abelianM

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "abelian", "abelianY", "abelian_gen", "genM_join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cAA : abelian A.

Hypothesis

cAA

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "abelian" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_abelian_cent : C \subset A -> A \subset 'C(C).

Proof. by move=> sCA; rewrite centsC (subset_trans sCA). Qed.

Lemma

sub_abelian_cent

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "centsC", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_abelian_cent2 : B \subset A -> C \subset A -> B \subset 'C(C).

Proof. by move=> sBA; move/sub_abelian_cent; apply: subset_trans. Qed.

Lemma

sub_abelian_cent2

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "sub_abelian_cent", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_abelian_norm : C \subset A -> A \subset 'N(C).

Proof. by move=> sCA; rewrite cents_norm ?sub_abelian_cent. Qed.

Lemma

sub_abelian_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cents_norm", "sub_abelian_cent" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_abelian_normal : (C \subset A) = (C <| A).

Proof. by rewrite /normal; case sHG: (C \subset A); rewrite // sub_abelian_norm. Qed.

Lemma

sub_abelian_normal

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normal", "sHG", "sub_abelian_norm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''N' ( A )"

:= (normaliser_group A) : Group_scope.

Notation

''N' ( A )

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normaliser_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C' ( A )"

:= (centraliser_group A) : Group_scope.

Notation

''C' ( A )

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "centraliser_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C' [ x ]"

:= (normaliser_group [set x%g]) : Group_scope.

Notation

''C' [ x ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normaliser_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''N_' G ( A )"

:= (setI_group G 'N(A)) : Group_scope.

Notation

''N_' G ( A )

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "setI_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' G ( A )"

:= (setI_group G 'C(A)) : Group_scope.

Notation

''C_' G ( A )

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "setI_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( G ) ( A )"

:= (setI_group G 'C(A)) (only parsing) : Group_scope.

Notation

''C_' ( G ) ( A )

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "setI_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' G [ x ]"

:= (setI_group G 'C[x]) : Group_scope.

Notation

''C_' G [ x ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "setI_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( G ) [ x ]"

:= (setI_group G 'C[x]) (only parsing) : Group_scope.

Notation

''C_' ( G ) [ x ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "setI_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxgroup A gP

:= maxset (fun A => group_set A && gP <<A>>%G) A.

Definition

maxgroup

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group_set", "maxset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mingroup A gP

:= minset (fun A => group_set A && gP <<A>>%G) A.

Definition

mingroup

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group_set", "minset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d