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
orbitJ G x : orbit 'J G x = x ^: G.	Proof. by []. Qed.	Lemma	orbitJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixJ A : 'Fix_('J)(A) = 'C(A).	Proof. apply/setP=> x; apply/afixP/centP=> cAx y Ay /=. by rewrite /commute conjgC cAx. by rewrite conjgE cAx ?mulKg. Qed.	Lemma	afixJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afixP", "apply", "centP", "commute", "conjgC", "conjgE", "mulKg", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabJ A : 'C(A \|'J) = 'C(A).	Proof. apply/setP=> x; apply/astabP/centP=> cAx y Ay /=. by apply: esym; rewrite conjgC cAx. by rewrite conjgE -cAx ?mulKg. Qed.	Lemma	astabJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabP", "centP", "conjgC", "conjgE", "mulKg", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1J x : 'C[x \|'J] = 'C[x].	Proof. by rewrite astabJ cent_set1. Qed.	Lemma	astab1J	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabJ", "cent_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsJ A : 'N(A \| 'J) = 'N(A).	Proof. by apply/setP=> x; rewrite -2!groupV !inE -conjg_preim -sub_conjg. Qed.	Lemma	astabsJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "conjg_preim", "groupV", "inE", "setP", "sub_conjg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setactJ A x : 'J^*%act A x = A :^ x.	Proof. by []. Qed.	Lemma	setactJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentJ A : 'C_(\|'J)(A) = 'C(A).	Proof. by rewrite gacentE ?setTI ?subsetT ?afixJ. Qed.	Lemma	gacentJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afixJ", "gacentE", "setTI", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbitRs G A : orbit 'Rs G A = rcosets A G.	Proof. by []. Qed.	Lemma	orbitRs	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit", "rcosets" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_afixRs_norms G x A : (G :* x \in 'Fix_('Rs)(A)) = (A \subset G :^ x).	Proof. rewrite inE /=; apply: eq_subset_r => a. rewrite inE rcosetE -(can2_eq (rcosetKV x) (rcosetK x)) -!rcosetM. rewrite eqEcard card_rcoset leqnn andbT mulgA (conjgCV x) mulgK. by rewrite -{2 3}(mulGid G) mulGS sub1set -mem_conjg. Qed.	Lemma	sub_afixRs_norms	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "can2_eq", "card_rcoset", "conjgCV", "eqEcard", "eq_subset_r", "inE", "leqnn", "mem_conjg", "mulGS", "mulGid", "mulgA", "mulgK", "rcosetE", "rcosetK", "rcosetKV", "rcosetM", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_afixRs_norm G x : (G :* x \in 'Fix_('Rs)(G)) = (x \in 'N(G)).	Proof. by rewrite sub_afixRs_norms -groupV inE sub_conjgV. Qed.	Lemma	sub_afixRs_norm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "groupV", "inE", "sub_afixRs_norms", "sub_conjgV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixRs_rcosets A G : 'Fix_(rcosets G A \| 'Rs)(G) = rcosets G 'N_A(G).	Proof. apply/setP=> Gx; apply/setIP/rcosetsP=> [[/rcosetsP[x Ax ->]]\|[x]]. by rewrite sub_afixRs_norm => Nx; exists x; rewrite // inE Ax. by case/setIP=> Ax Nx ->; rewrite -{1}rcosetE imset_f // sub_afixRs_norm. Qed.	Lemma	afixRs_rcosets	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "imset_f", "inE", "rcosetE", "rcosets", "rcosetsP", "setIP", "setP", "sub_afixRs_norm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1Rs G : 'C[G : {set gT} \| 'Rs] = G.	Proof. apply/setP=> x. by apply/astab1P/idP=> /= [<- \| Gx]; rewrite rcosetE ?rcoset_refl ?rcoset_id. Qed.	Lemma	astab1Rs	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astab1P", "gT", "rcosetE", "rcoset_id", "rcoset_refl", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actsRs_rcosets H G : [acts G, on rcosets H G \| 'Rs].	Proof. by rewrite -orbitRs acts_orbit ?subsetT. Qed.	Lemma	actsRs_rcosets	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_orbit", "on", "orbitRs", "rcosets", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
transRs_rcosets H G : [transitive G, on rcosets H G \| 'Rs].	Proof. by rewrite -orbitRs atrans_orbit. Qed.	Lemma	transRs_rcosets	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "atrans_orbit", "on", "orbitRs", "rcosets" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabRs_rcosets H G : 'C(rcosets H G \| 'Rs) = gcore H G.	Proof. have transGH := transRs_rcosets H G. by rewrite (astab_trans_gcore transGH (orbit_refl _ G _)) astab1Rs. Qed.	Lemma	astabRs_rcosets	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astab1Rs", "astab_trans_gcore", "gcore", "orbit_refl", "rcosets", "transRs_rcosets" ]	This is the second part of Aschbacher (5.7)	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbitJs G A : orbit 'Js G A = A :^: G.	Proof. by []. Qed.	Lemma	orbitJs	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1Js A : 'C[A \| 'Js] = 'N(A).	Proof. by apply/setP=> x; apply/astab1P/normP. Qed.	Lemma	astab1Js	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astab1P", "normP", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_conjugates A G : #\|A :^: G\| = #\|G : 'N_G(A)\|.	Proof. by rewrite card_orbit astab1Js. Qed.	Lemma	card_conjugates	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astab1Js", "card_orbit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixJG G A : (G \in 'Fix_('JG)(A)) = (A \subset 'N(G)).	Proof. by apply/afixP/normsP=> nG x Ax; apply/eqP; move/eqP: (nG x Ax). Qed.	Lemma	afixJG	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afixP", "apply", "nG", "normsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1JG G : 'C[G \| 'JG] = 'N(G).	Proof. by apply/setP=> x; apply/astab1P/normP=> [/congr_group \| /group_inj]. Qed.	Lemma	astab1JG	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astab1P", "congr_group", "group_inj", "normP", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dom_qactJ H : qact_dom 'J H = 'N(H).	Proof. by rewrite qact_domE ?subsetT ?astabsJ. Qed.	Lemma	dom_qactJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabsJ", "qact_dom", "qact_domE", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qactJ H (Hy : coset_of H) x : 'Q%act Hy x = if x \in 'N(H) then Hy ^ coset H x else Hy.	Proof. case: (cosetP Hy) => y Ny ->{Hy}. by rewrite qactEcond // dom_qactJ; case Nx: (x \in 'N(H)); rewrite ?morphJ. Qed.	Lemma	qactJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act", "coset", "cosetP", "coset_of", "dom_qactJ", "morphJ", "qactEcond" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actsQ A B H : A \subset 'N(H) -> A \subset 'N(B) -> [acts A, on B / H \| 'Q].	Proof. by move=> nHA nBA; rewrite acts_quotient // subsetI dom_qactJ nHA astabsJ. Qed.	Lemma	actsQ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_quotient", "astabsJ", "dom_qactJ", "nBA", "on", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsQ G H : H <\| G -> 'N(G / H \| 'Q) = 'N(H) :&: 'N(G).	Proof. by move=> nsHG; rewrite astabs_quotient // dom_qactJ astabsJ. Qed.	Lemma	astabsQ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabsJ", "astabs_quotient", "dom_qactJ", "nsHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabQ H Abar : 'C(Abar \|'Q) = coset H @*^-1 'C(Abar).	Proof. apply/setP=> x; rewrite inE /= dom_qactJ morphpreE in_setI /=. apply: andb_id2l => Nx; rewrite !inE -sub1set centsC cent_set1. apply: eq_subset_r => {Abar} Hy; rewrite inE qactJ Nx (sameP eqP conjg_fixP). by rewrite (sameP cent1P eqP) (sameP commgP eqP). Qed.	Lemma	astabQ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "cent1P", "cent_set1", "centsC", "commgP", "conjg_fixP", "coset", "dom_qactJ", "eq_subset_r", "inE", "in_setI", "morphpreE", "qactJ", "setP", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_astabQ A H Bbar : (A \subset 'C(Bbar \| 'Q)) = (A \subset 'N(H)) && (A / H \subset 'C(Bbar)).	Proof. rewrite astabQ -morphpreIdom subsetI; apply: andb_id2l => nHA. by rewrite -sub_quotient_pre. Qed.	Lemma	sub_astabQ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabQ", "morphpreIdom", "sub_quotient_pre", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_astabQR A B H : A \subset 'N(H) -> B \subset 'N(H) -> (A \subset 'C(B / H \| 'Q)) = ([~: A, B] \subset H).	Proof. move=> nHA nHB; rewrite sub_astabQ nHA /= (sameP commG1P eqP). by rewrite eqEsubset sub1G andbT -quotientR // quotient_sub1 // comm_subG. Qed.	Lemma	sub_astabQR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "commG1P", "comm_subG", "eqEsubset", "quotientR", "quotient_sub1", "sub1G", "sub_astabQ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabQR A H : A \subset 'N(H) -> 'C(A / H \| 'Q) = [set x in 'N(H) \| [~: [set x], A] \subset H].	Proof. move=> nHA; apply/setP=> x; rewrite astabQ -morphpreIdom 2!inE -astabQ. by case nHx: (x \in _); rewrite //= -sub1set sub_astabQR ?sub1set. Qed.	Lemma	astabQR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabQ", "inE", "morphpreIdom", "setP", "sub1set", "sub_astabQR" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_astabQ H Abar : 'C(Abar \| 'Q) / H = 'C(Abar).	Proof. by rewrite astabQ cosetpreK. Qed.	Lemma	quotient_astabQ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabQ", "cosetpreK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj_astabQ A H x : x \in 'N(H) -> 'C(A / H \| 'Q) :^ x = 'C(A :^ x / H \| 'Q).	Proof. move=> nHx; apply/setP=> y; rewrite !astabQ mem_conjg !in_setI -mem_conjg. rewrite -normJ (normP nHx) quotientJ //; apply/andb_id2l => nHy. by rewrite !inE centJ morphJ ?groupV ?morphV // -mem_conjg. Qed.	Lemma	conj_astabQ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabQ", "centJ", "groupV", "inE", "in_setI", "mem_conjg", "morphJ", "morphV", "normJ", "normP", "quotientJ", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index_cent1 x : #\|G : 'C_G[x]\| = #\|x ^: G\|.	Proof. by rewrite -astab1J -card_orbit. Qed.	Lemma	index_cent1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astab1J", "card_orbit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classes_partition : partition (classes G) G.	Proof. by apply: orbit_partition; apply/actsP=> x Gx y; apply: groupJr. Qed.	Lemma	classes_partition	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actsP", "apply", "classes", "groupJr", "orbit_partition", "partition" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_card_class : \sum_(C in classes G) #\|C\| = #\|G\|.	Proof. by apply: acts_sum_card_orbit; apply/actsP=> x Gx y; apply: groupJr. Qed.	Lemma	sum_card_class	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actsP", "acts_sum_card_orbit", "apply", "classes", "groupJr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_formula : \sum_(C in classes G) #\|G : 'C_G[repr C]\| = #\|G\|.	Proof. rewrite -sum_card_class; apply: eq_bigr => _ /imsetP[x Gx ->]. have: x \in x ^: G by rewrite -{1}(conjg1 x) imset_f. by case/mem_repr/imsetP=> y Gy ->; rewrite index_cent1 classGidl. Qed.	Lemma	class_formula	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "classGidl", "classes", "conjg1", "eq_bigr", "imsetP", "imset_f", "index_cent1", "mem_repr", "repr", "sum_card_class" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_classP : reflect {in G, forall x, x ^: G = [set x]} (abelian G).	Proof. rewrite /abelian -astabJ astabC. by apply: (iffP subsetP) => cGG x Gx; apply/orbit1P; apply: cGG. Qed.	Lemma	abelian_classP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "abelian", "apply", "astabC", "astabJ", "cGG", "orbit1P", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_classes_abelian : abelian G = (#\|classes G\| == #\|G\|).	Proof. have cGgt0 C: C \in classes G -> 1 <= #\|C\| ?= iff (#\|C\| == 1)%N. by case/imsetP=> x _ ->; rewrite eq_sym -index_cent1. rewrite -sum_card_class -sum1_card (leqif_sum cGgt0). apply/abelian_classP/forall_inP=> [cGG _ /imsetP[x Gx ->]\| cGG x Gx]. by rewrite cGG ?cards1. apply/esym/eqP; rewrite eqEcard sub1set ca...	Lemma	card_classes_abelian	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "abelian", "abelian_classP", "apply", "cGG", "cards1", "class_refl", "classes", "eqEcard", "eq_sym", "forall_inP", "imsetP", "imset_f", "index_cent1", "leq_eqVlt", "leqif_sum", "sub1set", "sum1_card", "sum_card_class" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentQ (gT : finGroupType) (H : {group gT}) (A : {set gT}) : 'C_(\|'Q)(A) = 'C(A / H).	Proof. apply/setP=> Hx; case: (cosetP Hx) => x Nx ->{Hx}. rewrite -sub_cent1 -astab1J astabC sub1set -(quotientInorm H A). have defD: qact_dom 'J H = 'N(H) by rewrite qact_domE ?subsetT ?astabsJ. rewrite !(inE, mem_quotient) //= defD setIC. apply/subsetP/subsetP=> [cAx _ /morphimP[a Na Aa ->] \| cAx a Aa]. by move/cAx...	Lemma	gacentQ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astab1J", "astabC", "astabsJ", "coset", "cosetP", "gT", "group", "inE", "mem_morphim", "mem_quotient", "morphJ", "morphimP", "qactE", "qact_dom", "qact_domE", "quotientInorm", "setIC", "setIP", "setP", "sub1set", "sub_cent1", "subsetP", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
autact	:= act ('P \ subsetT (Aut G)).	Definition	autact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "act", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aut_action	:= [action of autact].	Canonical	aut_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "action", "autact" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
autactK a : actperm aut_action a = a.	Proof. by apply/permP=> x; rewrite permE. Qed.	Lemma	autactK	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actperm", "apply", "aut_action", "permE", "permP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
autact_is_groupAction : is_groupAction G aut_action.	Proof. by move=> a Aa /=; rewrite autactK. Qed.	Lemma	autact_is_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aut_action", "autactK", "is_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aut_groupAction	:= GroupAction autact_is_groupAction.	Canonical	aut_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "autact_is_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Tp : prime #\|T\|.		Hypothesis	Tp	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cc : #[c]%g = #\|T\|.		Hypothesis	cc	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cp : prime #[c]%g.	Proof. by rewrite cc. Qed.	Let	cp	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "cc", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_prime_atrans : [transitive <[c]>, on setT \| 'P].	Proof. apply/imsetP; suff /existsP[x] : [exists x, ~~ (#\|orbit 'P <[c]> x\| < #[c])]. move=> oxT; suff /eqP orbit_x : orbit 'P <[c]> x == setT by exists x. by rewrite eqEcard subsetT cardsT -cc leqNgt. apply/forallP => olT; have o1 x : #\|orbit 'P <[c]> x\| == 1%N. by case/primeP: cp => _ /(_ _ (dvdn_orbit 'P _ x))/...	Lemma	perm_prime_atrans	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "c1", "card_orbit1", "cardsT", "cc", "cp", "cycle_id", "dvdn_orbit", "eqEcard", "existsP", "forallP", "imsetP", "leqNgt", "ltn_eqF", "mem_orbit", "on", "orbit", "order1", "perm1", "permP", "primeP", "set1P", "setT", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_prime_orbit x : orbit 'P <[c]> x = [set: T].	Proof. by apply: atransP => //; apply: perm_prime_atrans. Qed.	Lemma	perm_prime_orbit	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "atransP", "orbit", "perm_prime_atrans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_prime_astab x : 'C_<[c]>[x \| 'P]%g = 1%g.	Proof. by apply/card1_trivg/eqP; rewrite -(@eqn_pmul2l #\|orbit 'P <[c]> x\|) ?card_orbit_stab ?perm_prime_orbit ?cardsT ?muln1 ?prime_gt0// -cc. Qed.	Lemma	perm_prime_astab	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "card1_trivg", "card_orbit_stab", "cardsT", "cc", "eqn_pmul2l", "muln1", "orbit", "perm_prime_orbit", "prime_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Aut' G ]"	:= (aut_action G) : action_scope.	Notation	[ 'Aut' G ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aut_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Aut' G ]"	:= (aut_groupAction G) : groupAction_scope.	Notation	[ 'Aut' G ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aut_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut A	:= [set a \| perm_on A a & morphic A a].	Definition	Aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "morphic", "perm_on" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_morphic A a : a \in Aut A -> morphic A a.	Proof. by case/setIdP. Qed.	Lemma	Aut_morphic	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "morphic", "setIdP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
out_Aut A a x : a \in Aut A -> x \notin A -> a x = x.	Proof. by case/setIdP=> Aa _; apply: out_perm. Qed.	Lemma	out_Aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "apply", "out_perm", "setIdP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_Aut A : {in Aut A &, forall a b, {in A, a =1 b} -> a = b}.	Proof. move=> a g Aa Ag /= eqag; apply/permP=> x. by have [/eqag // \| /out_Aut out] := boolP (x \in A); rewrite !out. Qed.	Lemma	eq_Aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "apply", "out_Aut", "permP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
autm A a (AutAa : a \in Aut A)	:= morphm (Aut_morphic AutAa).	Definition	autm	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "Aut_morphic", "morphm" ]	The morphism that is represented by a given element of Aut A.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
autmE A a (AutAa : a \in Aut A) : autm AutAa = a.	Proof. by []. Qed.	Lemma	autmE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "autm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
autm_morphism A a aM	:= Eval hnf in [morphism of @autm A a aM].	Canonical	autm_morphism	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "autm", "morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_group_set : group_set (Aut G).	Proof. apply/group_setP; split=> [\|a b]. by rewrite inE perm_on1; apply/morphicP=> ? *; rewrite !permE. rewrite !inE => /andP[Ga aM] /andP[Gb bM]; rewrite perm_onM //=. apply/morphicP=> x y Gx Gy; rewrite !permM (morphicP aM) //. by rewrite (morphicP bM) ?perm_closed. Qed.	Lemma	Aut_group_set	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "apply", "group_set", "group_setP", "inE", "morphicP", "permE", "permM", "perm_closed", "perm_on1", "perm_onM", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_group	:= group Aut_group_set.	Canonical	Aut_group	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut_group_set", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
f	:= (autm AutGa).	Notation	f	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "autm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fE	:= (autmE AutGa).	Notation	fE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "autmE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_autm : 'injm f.	Proof. by apply/injmP; apply: in2W; apply: perm_inj. Qed.	Lemma	injm_autm	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "injmP", "perm_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_autm : 'ker f = 1.	Proof. by move/trivgP: injm_autm. Qed.	Lemma	ker_autm	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "injm_autm", "ker", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_autm : f @* G = G.	Proof. apply/setP=> x; rewrite morphimEdom (can_imset_pre _ (permK a)) inE. by have /[1!inE] /andP[/perm_closed <-] := AutGa; rewrite permKV. Qed.	Lemma	im_autm	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "can_imset_pre", "inE", "morphimEdom", "permK", "permKV", "perm_closed", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_closed x : x \in G -> a x \in G.	Proof. by move=> Gx; rewrite -im_autm; apply: mem_morphim. Qed.	Lemma	Aut_closed	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "im_autm", "mem_morphim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut1 : Aut 1 = 1.	Proof. apply/trivgP/subsetP=> a /= AutGa; apply/set1P. apply: eq_Aut (AutGa) (group1 _) _ => _ /set1P->. by rewrite -(autmE AutGa) morph1 perm1. Qed.	Lemma	Aut1	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "apply", "autmE", "eq_Aut", "group1", "morph1", "perm1", "set1P", "subsetP", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Aut' G ]"	:= (Aut_group G) (format "[ 'Aut' G ]") : Group_scope.	Notation	[ 'Aut' G ]	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Aut' G ]"	:= (Aut G) (only parsing) : group_scope.	Notation	[ 'Aut' G ]	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(injf : {in A &, injective f}) (sBf : f @: A \subset A).		Hypotheses	injf	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_in_inj : injective (fun x => if x \in A then f x else x).	Proof. move=> x y /=; wlog Ay: x y / y \in A. by move=> IH eqfxy; case: ifP (eqfxy); [symmetry \| case: ifP => //]; auto. rewrite Ay; case: ifP => [Ax \| nAx def_x]; first exact: injf. by case/negP: nAx; rewrite def_x (subsetP sBf) ?imset_f. Qed.	Lemma	perm_in_inj	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "imset_f", "injf", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_in	:= perm perm_in_inj.	Definition	perm_in	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "perm_in_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_in_on : perm_on A perm_in.	Proof. by apply/subsetP=> x; rewrite inE /= permE; case: ifP => // _; case/eqP. Qed.	Lemma	perm_in_on	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "inE", "permE", "perm_in", "perm_on", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_inE : {in A, perm_in =1 f}.	Proof. by move=> x Ax; rewrite /= permE Ax. Qed.	Lemma	perm_inE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "permE", "perm_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injf : 'injm f.		Hypothesis	injf	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_fixP A : A \subset G -> reflect (f @* A = A) (f @* A \subset A).	Proof. rewrite /morphim => sAG; have:= eqEcard (f @: A) A. rewrite (setIidPr sAG) card_in_imset ?leqnn ?andbT => [\|<-]; last exact: eqP. by move/injmP: injf; apply: sub_in2; apply/subsetP. Qed.	Lemma	morphim_fixP	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "card_in_imset", "eqEcard", "injf", "injmP", "last", "leqnn", "morphim", "sAG", "setIidPr", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Gf : f @* G = G.		Hypothesis	Gf	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aut_closed : f @: G \subset G.	Proof. by rewrite -morphimEdom; apply/morphim_fixP. Qed.	Lemma	aut_closed	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "morphimEdom", "morphim_fixP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aut	:= perm_in (injmP injf) aut_closed.	Definition	aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "aut_closed", "injf", "injmP", "perm_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
autE : {in G, aut =1 f}.	Proof. exact: perm_inE. Qed.	Lemma	autE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "aut", "perm_inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphic_aut : morphic G aut.	Proof. by apply/morphicP=> x y Gx Gy /=; rewrite !autE ?groupM // morphM. Qed.	Lemma	morphic_aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "aut", "autE", "groupM", "morphM", "morphic", "morphicP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_aut : aut \in Aut G.	Proof. by rewrite inE morphic_aut perm_in_on. Qed.	Lemma	Aut_aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "aut", "inE", "morphic_aut", "perm_in_on" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
imset_autE A : A \subset G -> aut @: A = f @* A.	Proof. move=> sAG; rewrite /morphim (setIidPr sAG). by apply: eq_in_imset; apply: sub_in1 autE; apply/subsetP. Qed.	Lemma	imset_autE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "aut", "autE", "eq_in_imset", "morphim", "sAG", "setIidPr", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
preim_autE A : A \subset G -> aut @^-1: A = f @*^-1 A.	Proof. move=> sAG; apply/setP=> x; rewrite !inE permE /=. by case Gx: (x \in G) => //; apply/negP=> Ax; rewrite (subsetP sAG) in Gx. Qed.	Lemma	preim_autE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "aut", "inE", "permE", "sAG", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(injf : 'injm f) (sGD : G \subset D).		Hypotheses	injf	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "sGD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
domG	:= subsetP sGD.	Let	domG	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "sGD", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_isom_subproof a : {a' \| a' \in Aut (f @* G) & a \in Aut G -> {in G, a' \o f =1 f \o a}}.	Proof. set Aut_a := autm (subgP (subg [Aut G] a)). have aDom: 'dom (f \o Aut_a \o invm injf) = f @* G. rewrite /dom /= morphpre_invm -morphpreIim; congr (f @* _). by rewrite [_ :&: D](setIidPl _) ?injmK ?injm_autm ?im_autm. have [af [def_af ker_af _ im_af]] := domP _ aDom. have inj_a': 'injm af by rewrite ker_af !i...	Lemma	Aut_isom_subproof	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "Aut_aut", "Dx", "aut", "autE", "autm", "autmE", "dom", "domG", "domP", "im_autm", "injf", "injmK", "injm_autm", "injm_comp", "injm_invm", "invm", "invmE", "mem_morphim", "morphim_comp", "morphim_invm", "morphpreIim", "morphpre_invm", "setIidPl", "subg", "sub...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_isom a	:= s2val (Aut_isom_subproof a).	Definition	Aut_isom	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut_isom_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_Aut_isom a : Aut_isom a \in Aut (f @* G).	Proof. by rewrite /Aut_isom; case: (Aut_isom_subproof a). Qed.	Lemma	Aut_Aut_isom	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "Aut_isom", "Aut_isom_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_isomE a : a \in Aut G -> {in G, forall x, Aut_isom a (f x) = f (a x)}.	Proof. by rewrite /Aut_isom; case: (Aut_isom_subproof a). Qed.	Lemma	Aut_isomE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "Aut_isom", "Aut_isom_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_isomM : {in Aut G &, {morph Aut_isom: x y / x * y}}.	Proof. move=> a b AutGa AutGb. apply: (eq_Aut (Aut_Aut_isom _)); rewrite ?groupM ?Aut_Aut_isom // => fx. case/morphimP=> x Dx Gx ->{fx}. by rewrite permM !Aut_isomE ?groupM /= ?permM ?Aut_closed. Qed.	Lemma	Aut_isomM	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "Aut_Aut_isom", "Aut_closed", "Aut_isom", "Aut_isomE", "Dx", "apply", "eq_Aut", "groupM", "morphimP", "permM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_isom_morphism	:= Morphism Aut_isomM.	Canonical	Aut_isom_morphism	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut_isomM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Aut_isom : 'injm Aut_isom.	Proof. apply/injmP=> a b AutGa AutGb eq_ab'; apply: (eq_Aut AutGa AutGb) => x Gx. by apply: (injmP injf); rewrite ?domG ?Aut_closed // -!Aut_isomE //= eq_ab'. Qed.	Lemma	injm_Aut_isom	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut_closed", "Aut_isom", "Aut_isomE", "apply", "domG", "eq_Aut", "injf", "injmP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_Aut_isom : Aut_isom injf sGD @* Aut G = Aut (f @* G).	Proof. apply/eqP; rewrite eqEcard; apply/andP; split. by apply/subsetP=> _ /morphimP[a _ AutGa ->]; apply: Aut_Aut_isom. have inj_isom' := injm_Aut_isom (injm_invm injf) (morphimS _ sGD). rewrite card_injm ?injm_Aut_isom // -(card_injm inj_isom') ?subset_leq_card //. apply/subsetP=> a /morphimP[a' _ AutfGa' def_a]. b...	Lemma	im_Aut_isom	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "Aut_Aut_isom", "Aut_isom", "apply", "card_injm", "eqEcard", "injf", "injm_Aut_isom", "injm_invm", "morphimP", "morphimS", "morphim_invm", "sGD", "split", "subsetP", "subset_leq_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_isomP : isom (Aut G) (Aut (f @* G)) (Aut_isom injf sGD).	Proof. by apply/isomP; split; [apply: injm_Aut_isom \| apply: im_Aut_isom]. Qed.	Lemma	Aut_isomP	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "Aut_isom", "apply", "im_Aut_isom", "injf", "injm_Aut_isom", "isom", "isomP", "sGD", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Aut : Aut (f @* G) \isog Aut G.	Proof. by rewrite isog_sym (isom_isog _ _ Aut_isomP). Qed.	Lemma	injm_Aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "Aut_isomP", "isog", "isog_sym", "isom_isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgm & {set gT}	:= fun x y : gT => y ^ x.	Definition	conjgm	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgmE A x y : conjgm A x y = y ^ x.	Proof. by []. Qed.	Lemma	conjgmE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "conjgm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgm_morphism A x	:= @Morphism _ _ A (conjgm A x) (in2W (fun y z => conjMg y z x)).	Canonical	conjgm_morphism	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "conjMg", "conjgm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_conj A x B : conjgm A x @* B = (A :&: B) :^ x.	Proof. by []. Qed.	Lemma	morphim_conj	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "conjgm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_conj x : 'injm (conjgm G x).	Proof. by apply/injmP; apply: in2W; apply: conjg_inj. Qed.	Lemma	injm_conj	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "conjg_inj", "conjgm", "injmP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbitJ G x : orbit 'J G x = x ^: G.

Proof. by []. Qed.

Lemma

orbitJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixJ A : 'Fix_('J)(A) = 'C(A).

Proof. apply/setP=> x; apply/afixP/centP=> cAx y Ay /=. by rewrite /commute conjgC cAx. by rewrite conjgE cAx ?mulKg. Qed.

Lemma

afixJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afixP", "apply", "centP", "commute", "conjgC", "conjgE", "mulKg", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabJ A : 'C(A |'J) = 'C(A).

Proof. apply/setP=> x; apply/astabP/centP=> cAx y Ay /=. by apply: esym; rewrite conjgC cAx. by rewrite conjgE -cAx ?mulKg. Qed.

Lemma

astabJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabP", "centP", "conjgC", "conjgE", "mulKg", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1J x : 'C[x |'J] = 'C[x].

Proof. by rewrite astabJ cent_set1. Qed.

Lemma

astab1J

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabJ", "cent_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsJ A : 'N(A | 'J) = 'N(A).

Proof. by apply/setP=> x; rewrite -2!groupV !inE -conjg_preim -sub_conjg. Qed.

Lemma

astabsJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "conjg_preim", "groupV", "inE", "setP", "sub_conjg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setactJ A x : 'J^*%act A x = A :^ x.

Proof. by []. Qed.

Lemma

setactJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "act" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentJ A : 'C_(|'J)(A) = 'C(A).

Proof. by rewrite gacentE ?setTI ?subsetT ?afixJ. Qed.

Lemma

gacentJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afixJ", "gacentE", "setTI", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbitRs G A : orbit 'Rs G A = rcosets A G.

Proof. by []. Qed.

Lemma

orbitRs

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit", "rcosets" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_afixRs_norms G x A : (G :* x \in 'Fix_('Rs)(A)) = (A \subset G :^ x).

Proof. rewrite inE /=; apply: eq_subset_r => a. rewrite inE rcosetE -(can2_eq (rcosetKV x) (rcosetK x)) -!rcosetM. rewrite eqEcard card_rcoset leqnn andbT mulgA (conjgCV x) mulgK. by rewrite -{2 3}(mulGid G) mulGS sub1set -mem_conjg. Qed.

Lemma

sub_afixRs_norms

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "can2_eq", "card_rcoset", "conjgCV", "eqEcard", "eq_subset_r", "inE", "leqnn", "mem_conjg", "mulGS", "mulGid", "mulgA", "mulgK", "rcosetE", "rcosetK", "rcosetKV", "rcosetM", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_afixRs_norm G x : (G :* x \in 'Fix_('Rs)(G)) = (x \in 'N(G)).

Proof. by rewrite sub_afixRs_norms -groupV inE sub_conjgV. Qed.

Lemma

sub_afixRs_norm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "groupV", "inE", "sub_afixRs_norms", "sub_conjgV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixRs_rcosets A G : 'Fix_(rcosets G A | 'Rs)(G) = rcosets G 'N_A(G).

Proof. apply/setP=> Gx; apply/setIP/rcosetsP=> [[/rcosetsP[x Ax ->]]|[x]]. by rewrite sub_afixRs_norm => Nx; exists x; rewrite // inE Ax. by case/setIP=> Ax Nx ->; rewrite -{1}rcosetE imset_f // sub_afixRs_norm. Qed.

Lemma

afixRs_rcosets

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "imset_f", "inE", "rcosetE", "rcosets", "rcosetsP", "setIP", "setP", "sub_afixRs_norm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1Rs G : 'C[G : {set gT} | 'Rs] = G.

Proof. apply/setP=> x. by apply/astab1P/idP=> /= [<- | Gx]; rewrite rcosetE ?rcoset_refl ?rcoset_id. Qed.

Lemma

astab1Rs

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astab1P", "gT", "rcosetE", "rcoset_id", "rcoset_refl", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actsRs_rcosets H G : [acts G, on rcosets H G | 'Rs].

Proof. by rewrite -orbitRs acts_orbit ?subsetT. Qed.

Lemma

actsRs_rcosets

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_orbit", "on", "orbitRs", "rcosets", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

transRs_rcosets H G : [transitive G, on rcosets H G | 'Rs].

Proof. by rewrite -orbitRs atrans_orbit. Qed.

Lemma

transRs_rcosets

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "atrans_orbit", "on", "orbitRs", "rcosets" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabRs_rcosets H G : 'C(rcosets H G | 'Rs) = gcore H G.

Proof. have transGH := transRs_rcosets H G. by rewrite (astab_trans_gcore transGH (orbit_refl _ G _)) astab1Rs. Qed.

Lemma

astabRs_rcosets

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astab1Rs", "astab_trans_gcore", "gcore", "orbit_refl", "rcosets", "transRs_rcosets" ]

This is the second part of Aschbacher (5.7)

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbitJs G A : orbit 'Js G A = A :^: G.

Proof. by []. Qed.

Lemma

orbitJs

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1Js A : 'C[A | 'Js] = 'N(A).

Proof. by apply/setP=> x; apply/astab1P/normP. Qed.

Lemma

astab1Js

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astab1P", "normP", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_conjugates A G : #|A :^: G| = #|G : 'N_G(A)|.

Proof. by rewrite card_orbit astab1Js. Qed.

Lemma

card_conjugates

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astab1Js", "card_orbit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixJG G A : (G \in 'Fix_('JG)(A)) = (A \subset 'N(G)).

Proof. by apply/afixP/normsP=> nG x Ax; apply/eqP; move/eqP: (nG x Ax). Qed.

Lemma

afixJG

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afixP", "apply", "nG", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1JG G : 'C[G | 'JG] = 'N(G).

Proof. by apply/setP=> x; apply/astab1P/normP=> [/congr_group | /group_inj]. Qed.

Lemma

astab1JG

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astab1P", "congr_group", "group_inj", "normP", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dom_qactJ H : qact_dom 'J H = 'N(H).

Proof. by rewrite qact_domE ?subsetT ?astabsJ. Qed.

Lemma

dom_qactJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabsJ", "qact_dom", "qact_domE", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qactJ H (Hy : coset_of H) x : 'Q%act Hy x = if x \in 'N(H) then Hy ^ coset H x else Hy.

Proof. case: (cosetP Hy) => y Ny ->{Hy}. by rewrite qactEcond // dom_qactJ; case Nx: (x \in 'N(H)); rewrite ?morphJ. Qed.

Lemma

qactJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "act", "coset", "cosetP", "coset_of", "dom_qactJ", "morphJ", "qactEcond" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actsQ A B H : A \subset 'N(H) -> A \subset 'N(B) -> [acts A, on B / H | 'Q].

Proof. by move=> nHA nBA; rewrite acts_quotient // subsetI dom_qactJ nHA astabsJ. Qed.

Lemma

actsQ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_quotient", "astabsJ", "dom_qactJ", "nBA", "on", "subsetI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsQ G H : H <| G -> 'N(G / H | 'Q) = 'N(H) :&: 'N(G).

Proof. by move=> nsHG; rewrite astabs_quotient // dom_qactJ astabsJ. Qed.

Lemma

astabsQ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabsJ", "astabs_quotient", "dom_qactJ", "nsHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabQ H Abar : 'C(Abar |'Q) = coset H @*^-1 'C(Abar).

Proof. apply/setP=> x; rewrite inE /= dom_qactJ morphpreE in_setI /=. apply: andb_id2l => Nx; rewrite !inE -sub1set centsC cent_set1. apply: eq_subset_r => {Abar} Hy; rewrite inE qactJ Nx (sameP eqP conjg_fixP). by rewrite (sameP cent1P eqP) (sameP commgP eqP). Qed.

Lemma

astabQ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "cent1P", "cent_set1", "centsC", "commgP", "conjg_fixP", "coset", "dom_qactJ", "eq_subset_r", "inE", "in_setI", "morphpreE", "qactJ", "setP", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_astabQ A H Bbar : (A \subset 'C(Bbar | 'Q)) = (A \subset 'N(H)) && (A / H \subset 'C(Bbar)).

Proof. rewrite astabQ -morphpreIdom subsetI; apply: andb_id2l => nHA. by rewrite -sub_quotient_pre. Qed.

Lemma

sub_astabQ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabQ", "morphpreIdom", "sub_quotient_pre", "subsetI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_astabQR A B H : A \subset 'N(H) -> B \subset 'N(H) -> (A \subset 'C(B / H | 'Q)) = ([~: A, B] \subset H).

Proof. move=> nHA nHB; rewrite sub_astabQ nHA /= (sameP commG1P eqP). by rewrite eqEsubset sub1G andbT -quotientR // quotient_sub1 // comm_subG. Qed.

Lemma

sub_astabQR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "commG1P", "comm_subG", "eqEsubset", "quotientR", "quotient_sub1", "sub1G", "sub_astabQ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabQR A H : A \subset 'N(H) -> 'C(A / H | 'Q) = [set x in 'N(H) | [~: [set x], A] \subset H].

Proof. move=> nHA; apply/setP=> x; rewrite astabQ -morphpreIdom 2!inE -astabQ. by case nHx: (x \in _); rewrite //= -sub1set sub_astabQR ?sub1set. Qed.

Lemma

astabQR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabQ", "inE", "morphpreIdom", "setP", "sub1set", "sub_astabQR" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_astabQ H Abar : 'C(Abar | 'Q) / H = 'C(Abar).

Proof. by rewrite astabQ cosetpreK. Qed.

Lemma

quotient_astabQ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabQ", "cosetpreK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj_astabQ A H x : x \in 'N(H) -> 'C(A / H | 'Q) :^ x = 'C(A :^ x / H | 'Q).

Proof. move=> nHx; apply/setP=> y; rewrite !astabQ mem_conjg !in_setI -mem_conjg. rewrite -normJ (normP nHx) quotientJ //; apply/andb_id2l => nHy. by rewrite !inE centJ morphJ ?groupV ?morphV // -mem_conjg. Qed.

Lemma

conj_astabQ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabQ", "centJ", "groupV", "inE", "in_setI", "mem_conjg", "morphJ", "morphV", "normJ", "normP", "quotientJ", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

index_cent1 x : #|G : 'C_G[x]| = #|x ^: G|.

Proof. by rewrite -astab1J -card_orbit. Qed.

Lemma

index_cent1

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astab1J", "card_orbit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classes_partition : partition (classes G) G.

Proof. by apply: orbit_partition; apply/actsP=> x Gx y; apply: groupJr. Qed.

Lemma

classes_partition

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actsP", "apply", "classes", "groupJr", "orbit_partition", "partition" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_card_class : \sum_(C in classes G) #|C| = #|G|.

Proof. by apply: acts_sum_card_orbit; apply/actsP=> x Gx y; apply: groupJr. Qed.

Lemma

sum_card_class

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actsP", "acts_sum_card_orbit", "apply", "classes", "groupJr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_formula : \sum_(C in classes G) #|G : 'C_G[repr C]| = #|G|.

Proof. rewrite -sum_card_class; apply: eq_bigr => _ /imsetP[x Gx ->]. have: x \in x ^: G by rewrite -{1}(conjg1 x) imset_f. by case/mem_repr/imsetP=> y Gy ->; rewrite index_cent1 classGidl. Qed.

Lemma

class_formula

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "classGidl", "classes", "conjg1", "eq_bigr", "imsetP", "imset_f", "index_cent1", "mem_repr", "repr", "sum_card_class" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_classP : reflect {in G, forall x, x ^: G = [set x]} (abelian G).

Proof. rewrite /abelian -astabJ astabC. by apply: (iffP subsetP) => cGG x Gx; apply/orbit1P; apply: cGG. Qed.

Lemma

abelian_classP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "abelian", "apply", "astabC", "astabJ", "cGG", "orbit1P", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_classes_abelian : abelian G = (#|classes G| == #|G|).

Proof. have cGgt0 C: C \in classes G -> 1 <= #|C| ?= iff (#|C| == 1)%N. by case/imsetP=> x _ ->; rewrite eq_sym -index_cent1. rewrite -sum_card_class -sum1_card (leqif_sum cGgt0). apply/abelian_classP/forall_inP=> [cGG _ /imsetP[x Gx ->]| cGG x Gx]. by rewrite cGG ?cards1. apply/esym/eqP; rewrite eqEcard sub1set ca...

Lemma

card_classes_abelian

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "abelian", "abelian_classP", "apply", "cGG", "cards1", "class_refl", "classes", "eqEcard", "eq_sym", "forall_inP", "imsetP", "imset_f", "index_cent1", "leq_eqVlt", "leqif_sum", "sub1set", "sum1_card", "sum_card_class" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentQ (gT : finGroupType) (H : {group gT}) (A : {set gT}) : 'C_(|'Q)(A) = 'C(A / H).

Proof. apply/setP=> Hx; case: (cosetP Hx) => x Nx ->{Hx}. rewrite -sub_cent1 -astab1J astabC sub1set -(quotientInorm H A). have defD: qact_dom 'J H = 'N(H) by rewrite qact_domE ?subsetT ?astabsJ. rewrite !(inE, mem_quotient) //= defD setIC. apply/subsetP/subsetP=> [cAx _ /morphimP[a Na Aa ->] | cAx a Aa]. by move/cAx...

Lemma

gacentQ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astab1J", "astabC", "astabsJ", "coset", "cosetP", "gT", "group", "inE", "mem_morphim", "mem_quotient", "morphJ", "morphimP", "qactE", "qact_dom", "qact_domE", "quotientInorm", "setIC", "setIP", "setP", "sub1set", "sub_cent1", "subsetP", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

autact

:= act ('P \ subsetT (Aut G)).

Definition

autact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "act", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aut_action

:= [action of autact].

Canonical

aut_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "action", "autact" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

autactK a : actperm aut_action a = a.

Proof. by apply/permP=> x; rewrite permE. Qed.

Lemma

autactK

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actperm", "apply", "aut_action", "permE", "permP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

autact_is_groupAction : is_groupAction G aut_action.

Proof. by move=> a Aa /=; rewrite autactK. Qed.

Lemma

autact_is_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aut_action", "autactK", "is_groupAction" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aut_groupAction

:= GroupAction autact_is_groupAction.

Canonical

aut_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "autact_is_groupAction" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Tp : prime #|T|.

Hypothesis

Tp

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "prime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cc : #[c]%g = #|T|.

Hypothesis

cc

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cp : prime #[c]%g.

Proof. by rewrite cc. Qed.

Let

cp

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "cc", "prime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_prime_atrans : [transitive <[c]>, on setT | 'P].

Proof. apply/imsetP; suff /existsP[x] : [exists x, ~~ (#|orbit 'P <[c]> x| < #[c])]. move=> oxT; suff /eqP orbit_x : orbit 'P <[c]> x == setT by exists x. by rewrite eqEcard subsetT cardsT -cc leqNgt. apply/forallP => olT; have o1 x : #|orbit 'P <[c]> x| == 1%N. by case/primeP: cp => _ /(_ _ (dvdn_orbit 'P _ x))/...

Lemma

perm_prime_atrans

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "c1", "card_orbit1", "cardsT", "cc", "cp", "cycle_id", "dvdn_orbit", "eqEcard", "existsP", "forallP", "imsetP", "leqNgt", "ltn_eqF", "mem_orbit", "on", "orbit", "order1", "perm1", "permP", "primeP", "set1P", "setT", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_prime_orbit x : orbit 'P <[c]> x = [set: T].

Proof. by apply: atransP => //; apply: perm_prime_atrans. Qed.

Lemma

perm_prime_orbit

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "atransP", "orbit", "perm_prime_atrans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_prime_astab x : 'C_<[c]>[x | 'P]%g = 1%g.

Proof. by apply/card1_trivg/eqP; rewrite -(@eqn_pmul2l #|orbit 'P <[c]> x|) ?card_orbit_stab ?perm_prime_orbit ?cardsT ?muln1 ?prime_gt0// -cc. Qed.

Lemma

perm_prime_astab

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "card1_trivg", "card_orbit_stab", "cardsT", "cc", "eqn_pmul2l", "muln1", "orbit", "perm_prime_orbit", "prime_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Aut' G ]"

:= (aut_action G) : action_scope.

Notation

[ 'Aut' G ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aut_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Aut' G ]"

:= (aut_groupAction G) : groupAction_scope.

Notation

[ 'Aut' G ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aut_groupAction" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut A

:= [set a | perm_on A a & morphic A a].

Definition

Aut

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "morphic", "perm_on" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_morphic A a : a \in Aut A -> morphic A a.

Proof. by case/setIdP. Qed.

Lemma

Aut_morphic

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "morphic", "setIdP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

out_Aut A a x : a \in Aut A -> x \notin A -> a x = x.

Proof. by case/setIdP=> Aa _; apply: out_perm. Qed.

Lemma

out_Aut

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "apply", "out_perm", "setIdP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_Aut A : {in Aut A &, forall a b, {in A, a =1 b} -> a = b}.

Proof. move=> a g Aa Ag /= eqag; apply/permP=> x. by have [/eqag // | /out_Aut out] := boolP (x \in A); rewrite !out. Qed.

Lemma

eq_Aut

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "apply", "out_Aut", "permP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

autm A a (AutAa : a \in Aut A)

:= morphm (Aut_morphic AutAa).

Definition

autm

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "Aut_morphic", "morphm" ]

The morphism that is represented by a given element of Aut A.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

autmE A a (AutAa : a \in Aut A) : autm AutAa = a.

Proof. by []. Qed.

Lemma

autmE

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "autm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

autm_morphism A a aM

:= Eval hnf in [morphism of @autm A a aM].

Canonical

autm_morphism

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "autm", "morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_group_set : group_set (Aut G).

Proof. apply/group_setP; split=> [|a b]. by rewrite inE perm_on1; apply/morphicP=> ? *; rewrite !permE. rewrite !inE => /andP[Ga aM] /andP[Gb bM]; rewrite perm_onM //=. apply/morphicP=> x y Gx Gy; rewrite !permM (morphicP aM) //. by rewrite (morphicP bM) ?perm_closed. Qed.

Lemma

Aut_group_set

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "apply", "group_set", "group_setP", "inE", "morphicP", "permE", "permM", "perm_closed", "perm_on1", "perm_onM", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_group

:= group Aut_group_set.

Canonical

Aut_group

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut_group_set", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

f

:= (autm AutGa).

Notation

f

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "autm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fE

:= (autmE AutGa).

Notation

fE

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "autmE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_autm : 'injm f.

Proof. by apply/injmP; apply: in2W; apply: perm_inj. Qed.

Lemma

injm_autm

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "injmP", "perm_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_autm : 'ker f = 1.

Proof. by move/trivgP: injm_autm. Qed.

Lemma

ker_autm

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "injm_autm", "ker", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_autm : f @* G = G.

Proof. apply/setP=> x; rewrite morphimEdom (can_imset_pre _ (permK a)) inE. by have /[1!inE] /andP[/perm_closed <-] := AutGa; rewrite permKV. Qed.

Lemma

im_autm

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "can_imset_pre", "inE", "morphimEdom", "permK", "permKV", "perm_closed", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_closed x : x \in G -> a x \in G.

Proof. by move=> Gx; rewrite -im_autm; apply: mem_morphim. Qed.

Lemma

Aut_closed

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "im_autm", "mem_morphim" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut1 : Aut 1 = 1.

Proof. apply/trivgP/subsetP=> a /= AutGa; apply/set1P. apply: eq_Aut (AutGa) (group1 _) _ => _ /set1P->. by rewrite -(autmE AutGa) morph1 perm1. Qed.

Lemma

Aut1

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "apply", "autmE", "eq_Aut", "group1", "morph1", "perm1", "set1P", "subsetP", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Aut' G ]"

:= (Aut_group G) (format "[ 'Aut' G ]") : Group_scope.

Notation

[ 'Aut' G ]

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Aut' G ]"

:= (Aut G) (only parsing) : group_scope.

Notation

[ 'Aut' G ]

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(injf : {in A &, injective f}) (sBf : f @: A \subset A).

Hypotheses

injf

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_in_inj : injective (fun x => if x \in A then f x else x).

Proof. move=> x y /=; wlog Ay: x y / y \in A. by move=> IH eqfxy; case: ifP (eqfxy); [symmetry | case: ifP => //]; auto. rewrite Ay; case: ifP => [Ax | nAx def_x]; first exact: injf. by case/negP: nAx; rewrite def_x (subsetP sBf) ?imset_f. Qed.

Lemma

perm_in_inj

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "imset_f", "injf", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_in

:= perm perm_in_inj.

Definition

perm_in

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "perm_in_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_in_on : perm_on A perm_in.

Proof. by apply/subsetP=> x; rewrite inE /= permE; case: ifP => // _; case/eqP. Qed.

Lemma

perm_in_on

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "inE", "permE", "perm_in", "perm_on", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_inE : {in A, perm_in =1 f}.

Proof. by move=> x Ax; rewrite /= permE Ax. Qed.

Lemma

perm_inE

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "permE", "perm_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injf : 'injm f.

Hypothesis

injf

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_fixP A : A \subset G -> reflect (f @* A = A) (f @* A \subset A).

Proof. rewrite /morphim => sAG; have:= eqEcard (f @: A) A. rewrite (setIidPr sAG) card_in_imset ?leqnn ?andbT => [|<-]; last exact: eqP. by move/injmP: injf; apply: sub_in2; apply/subsetP. Qed.

Lemma

morphim_fixP

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "card_in_imset", "eqEcard", "injf", "injmP", "last", "leqnn", "morphim", "sAG", "setIidPr", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Gf : f @* G = G.

Hypothesis

Gf

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aut_closed : f @: G \subset G.

Proof. by rewrite -morphimEdom; apply/morphim_fixP. Qed.

Lemma

aut_closed

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "morphimEdom", "morphim_fixP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aut

:= perm_in (injmP injf) aut_closed.

Definition

aut

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "aut_closed", "injf", "injmP", "perm_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

autE : {in G, aut =1 f}.

Proof. exact: perm_inE. Qed.

Lemma

autE

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "aut", "perm_inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphic_aut : morphic G aut.

Proof. by apply/morphicP=> x y Gx Gy /=; rewrite !autE ?groupM // morphM. Qed.

Lemma

morphic_aut

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "aut", "autE", "groupM", "morphM", "morphic", "morphicP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_aut : aut \in Aut G.

Proof. by rewrite inE morphic_aut perm_in_on. Qed.

Lemma

Aut_aut

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "aut", "inE", "morphic_aut", "perm_in_on" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

imset_autE A : A \subset G -> aut @: A = f @* A.

Proof. move=> sAG; rewrite /morphim (setIidPr sAG). by apply: eq_in_imset; apply: sub_in1 autE; apply/subsetP. Qed.

Lemma

imset_autE

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "aut", "autE", "eq_in_imset", "morphim", "sAG", "setIidPr", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

preim_autE A : A \subset G -> aut @^-1: A = f @*^-1 A.

Proof. move=> sAG; apply/setP=> x; rewrite !inE permE /=. by case Gx: (x \in G) => //; apply/negP=> Ax; rewrite (subsetP sAG) in Gx. Qed.

Lemma

preim_autE

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "aut", "inE", "permE", "sAG", "setP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(injf : 'injm f) (sGD : G \subset D).

Hypotheses

injf

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "sGD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

domG

:= subsetP sGD.

Let

domG

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "sGD", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_isom_subproof a : {a' | a' \in Aut (f @* G) & a \in Aut G -> {in G, a' \o f =1 f \o a}}.

Proof. set Aut_a := autm (subgP (subg [Aut G] a)). have aDom: 'dom (f \o Aut_a \o invm injf) = f @* G. rewrite /dom /= morphpre_invm -morphpreIim; congr (f @* _). by rewrite [_ :&: D](setIidPl _) ?injmK ?injm_autm ?im_autm. have [af [def_af ker_af _ im_af]] := domP _ aDom. have inj_a': 'injm af by rewrite ker_af !i...

Lemma

Aut_isom_subproof

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "Aut_aut", "Dx", "aut", "autE", "autm", "autmE", "dom", "domG", "domP", "im_autm", "injf", "injmK", "injm_autm", "injm_comp", "injm_invm", "invm", "invmE", "mem_morphim", "morphim_comp", "morphim_invm", "morphpreIim", "morphpre_invm", "setIidPl", "subg", "sub...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_isom a

:= s2val (Aut_isom_subproof a).

Definition

Aut_isom

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut_isom_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_Aut_isom a : Aut_isom a \in Aut (f @* G).

Proof. by rewrite /Aut_isom; case: (Aut_isom_subproof a). Qed.

Lemma

Aut_Aut_isom

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "Aut_isom", "Aut_isom_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_isomE a : a \in Aut G -> {in G, forall x, Aut_isom a (f x) = f (a x)}.

Proof. by rewrite /Aut_isom; case: (Aut_isom_subproof a). Qed.

Lemma

Aut_isomE

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "Aut_isom", "Aut_isom_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_isomM : {in Aut G &, {morph Aut_isom: x y / x * y}}.

Proof. move=> a b AutGa AutGb. apply: (eq_Aut (Aut_Aut_isom _)); rewrite ?groupM ?Aut_Aut_isom // => fx. case/morphimP=> x Dx Gx ->{fx}. by rewrite permM !Aut_isomE ?groupM /= ?permM ?Aut_closed. Qed.

Lemma

Aut_isomM

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "Aut_Aut_isom", "Aut_closed", "Aut_isom", "Aut_isomE", "Dx", "apply", "eq_Aut", "groupM", "morphimP", "permM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_isom_morphism

:= Morphism Aut_isomM.

Canonical

Aut_isom_morphism

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut_isomM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Aut_isom : 'injm Aut_isom.

Proof. apply/injmP=> a b AutGa AutGb eq_ab'; apply: (eq_Aut AutGa AutGb) => x Gx. by apply: (injmP injf); rewrite ?domG ?Aut_closed // -!Aut_isomE //= eq_ab'. Qed.

Lemma

injm_Aut_isom

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut_closed", "Aut_isom", "Aut_isomE", "apply", "domG", "eq_Aut", "injf", "injmP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_Aut_isom : Aut_isom injf sGD @* Aut G = Aut (f @* G).

Proof. apply/eqP; rewrite eqEcard; apply/andP; split. by apply/subsetP=> _ /morphimP[a _ AutGa ->]; apply: Aut_Aut_isom. have inj_isom' := injm_Aut_isom (injm_invm injf) (morphimS _ sGD). rewrite card_injm ?injm_Aut_isom // -(card_injm inj_isom') ?subset_leq_card //. apply/subsetP=> a /morphimP[a' _ AutfGa' def_a]. b...

Lemma

im_Aut_isom

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "Aut_Aut_isom", "Aut_isom", "apply", "card_injm", "eqEcard", "injf", "injm_Aut_isom", "injm_invm", "morphimP", "morphimS", "morphim_invm", "sGD", "split", "subsetP", "subset_leq_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_isomP : isom (Aut G) (Aut (f @* G)) (Aut_isom injf sGD).

Proof. by apply/isomP; split; [apply: injm_Aut_isom | apply: im_Aut_isom]. Qed.

Lemma

Aut_isomP

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "Aut_isom", "apply", "im_Aut_isom", "injf", "injm_Aut_isom", "isom", "isomP", "sGD", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Aut : Aut (f @* G) \isog Aut G.

Proof. by rewrite isog_sym (isom_isog _ _ Aut_isomP). Qed.

Lemma

injm_Aut

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "Aut", "Aut_isomP", "isog", "isog_sym", "isom_isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjgm & {set gT}

:= fun x y : gT => y ^ x.

Definition

conjgm

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjgmE A x y : conjgm A x y = y ^ x.

Proof. by []. Qed.

Lemma

conjgmE

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "conjgm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjgm_morphism A x

:= @Morphism _ _ A (conjgm A x) (in2W (fun y z => conjMg y z x)).

Canonical

conjgm_morphism

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "conjMg", "conjgm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_conj A x B : conjgm A x @* B = (A :&: B) :^ x.

Proof. by []. Qed.

Lemma

morphim_conj

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "conjgm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_conj x : 'injm (conjgm G x).

Proof. by apply/injmP; apply: in2W; apply: conjg_inj. Qed.

Lemma

injm_conj

finite_group

finite_group/automorphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]

[ "apply", "conjg_inj", "conjgm", "injmP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d