fact stringlengths 10 44.1k | type stringclasses 15
values | library stringclasses 1
value | imports listlengths 5 17 | filename stringclasses 34
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
p_constrained p (G : {set gT}) := forall P : {group gT}, p.-Sylow('O_{p^',p}(G)) P -> 'C_G(P) \subset 'O_{p^',p}(G). | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p_constrained | |
p_abelian_constrained p (G : {set gT}) := forall S A : {group gT}, p.-Sylow(G) S -> abelian A -> A <| S -> A \subset 'O_{p^',p}(G). | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p_abelian_constrained | |
p_stable p (G : {set gT}) := forall P A : {group gT}, p.-group P -> 'O_p^'(G) * P <| G -> p.-subgroup('N_G(P)) A -> [~: P, A, A] = 1 -> A / 'C_G(P) \subset 'O_p('N_G(P) / 'C_G(P)). | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p_stable | |
generated_by (gp : pred {group gT}) (E : {set gT}) := [exists gE : {set {group gT}}, <<\bigcup_(G in gE | gp G) G>> == E]. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | generated_by | |
norm_abelian (D : {set gT}) : pred {group gT} := fun A => (D \subset 'N(A)) && abelian A. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | norm_abelian | |
p_norm_abelian p (D : {set gT}) : pred {group gT} := fun A => p.-group A && norm_abelian D A. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p_norm_abelian | |
Puig_succ (D E : {set gT}) := <<\bigcup_(A in subgroups D | norm_abelian E A) A>>. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_succ | |
Puig_rec D := iter n (Puig_succ D) 1. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_rec | |
Puig_at := nosimpl Puig_rec. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_at | |
Puig_inf (gT : finGroupType) (G : {set gT}) := Puig_at #|G|.*2 G. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_inf | |
Puig (gT : finGroupType) (G : {set gT}) := Puig_at #|G|.*2.+1 G. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig | |
minnormal_solvable_abelem gT (M G : {group gT}) : minnormal M G -> solvable M -> is_abelem M. Proof. by move=> minM solM; case: (minnormal_solvable minM (subxx _) solM). Qed. (* This is B & G, Lemma 1.2, second part. *) | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | minnormal_solvable_abelem | |
minnormal_solvable_Fitting_center gT (M G : {group gT}) : minnormal M G -> M \subset G -> solvable M -> M \subset 'Z('F(G)). Proof. have nZG: 'Z('F(G)) <| G by rewrite !gFnormal_trans. move=> minM sMG solM; have[/andP[ntM nMG] minM'] := mingroupP minM. apply/setIidPl/minM'; last exact: subsetIl. apply/andP; split; last... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | minnormal_solvable_Fitting_center | |
sol_chief_abelem gT (G V U : {group gT}) : solvable G -> chief_factor G V U -> is_abelem (U / V). Proof. move=> solG chiefUV; have minUV := chief_factor_minnormal chiefUV. have [|//] := minnormal_solvable minUV (quotientS _ _) (quotient_sol _ solG). by case/and3P: chiefUV. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | sol_chief_abelem | |
sG'G : G' \subset G. Proof. exact: normal_sub. Qed. | Let | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | sG'G | |
nG'G : G \subset 'N(G'). Proof. exact: normal_norm. Qed. | Let | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | nG'G | |
nsF'G : 'F(G') <| G. Proof. exact: gFnormal_trans. Qed. | Let | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | nsF'G | |
Gchief (UV : {group gT} * {group gT}) := chief_factor G UV.2 UV.1. | Let | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Gchief | |
H := \bigcap_(UV | Gchief UV) 'C(UV.1 / UV.2 | 'Q). | Let | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | H | |
H' := G' :&: \bigcap_(UV | Gchief UV && (UV.1 \subset 'F(G'))) 'C(UV.1 / UV.2 | 'Q). (* This is B & G Proposition 1.2, non trivial inclusion of the first equality.*) | Let | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | H' | |
Fitting_stab_chief : 'F(G') \subset H. Proof. apply/bigcapsP=> [[U V] /= chiefUV]. have minUV: minnormal (U / V) (G / V) := chief_factor_minnormal chiefUV. have{chiefUV} [/=/maxgroupp/andP[_ nVG] sUG nUG] := and3P chiefUV. have solUV: solvable (U / V) by rewrite quotient_sol // (solvableS sUG). have{solUV minUV}: U / V... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Fitting_stab_chief | |
chief_stab_sub_Fitting : H' \subset 'F(G'). Proof. without loss: / {K | [min K | K <| G & ~~ (K \subset 'F(G'))] & K \subset H'}. move=> IH; apply: wlog_neg => s'H'F; apply/IH/mingroup_exists=> {IH}/=. rewrite /normal subIset ?sG'G ?normsI ?norms_bigcap {s'H'F}//. apply/bigcapsP=> /= U /andP[/and3P[/maxgroupp/andP/=[_ ... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | chief_stab_sub_Fitting | |
cent_sub_Fitting gT (G : {group gT}) : solvable G -> 'C_G('F(G)) \subset 'F(G). Proof. move=> solG; apply: subset_trans (chief_stab_sub_Fitting solG _) => //. rewrite subsetI subsetIl; apply/bigcapsP=> [[U V]] /=. case/andP=> /andP[/maxgroupp/andP[_ nVG] _] sUF. by rewrite astabQ (subset_trans _ (morphpre_cent _ _)) //... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | cent_sub_Fitting | |
coprime_trivg_cent_Fitting gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|G| #|A| -> solvable G -> 'C_A(G) = 1 -> 'C_A('F(G)) = 1. Proof. move=> nGA coGA solG regAG; without loss cycA: A nGA coGA regAG / cyclic A. move=> IH; apply/trivgP/subsetP=> a; rewrite -!cycle_subG subsetI. case/andP=> saA /setIidPl <-. rew... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_trivg_cent_Fitting | |
coprime_cent_Fitting gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|G| #|A| -> solvable G -> 'C_A('F(G)) \subset 'C(G). Proof. move=> nGA coGA solG; apply: subset_trans (subsetIr A _); set C := 'C_A(G). rewrite -quotient_sub1 /= -/C; last first. by rewrite subIset // normsI ?normG // norms_cent. apply: subset_tra... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_cent_Fitting | |
coprimeR_cent_prod gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|[~: G, A]| #|A| -> solvable [~: G, A] -> [~: G, A] * 'C_G(A) = G. Proof. move=> nGA coRA solR; apply/eqP; rewrite eqEsubset mulG_subG commg_subl nGA. rewrite subsetIl -quotientSK ?commg_norml //=. rewrite coprime_norm_quotient_cent ?commg_normr //=... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprimeR_cent_prod | |
coprime_cent_prod gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|G| #|A| -> solvable G -> [~: G, A] * 'C_G(A) = G. Proof. move=> nGA; have sRG: [~: G, A] \subset G by rewrite commg_subl. rewrite -(Lagrange sRG) coprimeMl => /andP[coRA _] /(solvableS sRG). exact: coprimeR_cent_prod. Qed. (* This is B & G, Proposit... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_cent_prod | |
coprime_commGid gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|G| #|A| -> solvable G -> [~: G, A, A] = [~: G, A]. Proof. move=> nGA coGA solG; apply/eqP; rewrite eqEsubset commSg ?commg_subl //. have nAC: 'C_G(A) \subset 'N(A) by rewrite subIset ?cent_sub ?orbT. rewrite -{1}(coprime_cent_prod nGA) // commMG //=; ... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_commGid | |
coprime_commGG1P gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|G| #|A| -> solvable G -> [~: G, A, A] = 1 -> A \subset 'C(G). Proof. by move=> nGA coGA solG; rewrite centsC coprime_commGid // => /commG1P. Qed. (* This is B & G, Proposition 1.6(d), TI-part, from finmod.v *) | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_commGG1P | |
coprime_abel_cent_TI := coprime_abel_cent_TI. (* This is B & G, Proposition 1.6(d) (direct product) *) | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_abel_cent_TI | |
coprime_abelian_cent_dprod gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|G| #|A| -> abelian G -> [~: G, A] \x 'C_G(A) = G. Proof. move=> nGA coGA abelG; rewrite dprodE ?coprime_cent_prod ?abelian_sol //. by rewrite subIset 1?(subset_trans abelG) // centS // commg_subl. by apply/trivgP; rewrite /= setICA coprime_... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_abelian_cent_dprod | |
coprime_abelian_faithful_Ohm1 gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|G| #|A| -> abelian G -> A \subset 'C('Ohm_1(G)) -> A \subset 'C(G). Proof. move=> nGA coGA cGG; rewrite !(centsC A) => cAG1. have /dprodP[_ defG _ tiRC] := coprime_abelian_cent_dprod nGA coGA cGG. have sRG: [~: G, A] \subset G by rewrite... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_abelian_faithful_Ohm1 | |
coprime_cent_Phi gT p (A G : {group gT}) : p.-group G -> coprime #|G| #|A| -> [~: G, A] \subset 'Phi(G) -> A \subset 'C(G). Proof. move=> pG coGA sRphi; rewrite centsC; apply/setIidPl. rewrite -['C_G(A)]genGid; apply/Phi_nongen/eqP. rewrite eqEsubset join_subG Phi_sub subsetIl -genM_join sub_gen //=. rewrite -{1}(copri... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_cent_Phi | |
stable_factor_cent gT (A G H : {group gT}) : A \subset 'C(H) -> stable_factor A H G -> coprime #|G| #|A| -> solvable G -> A \subset 'C(G). Proof. move=> cHA /and3P[sRH sHG nHG] coGA solG. suffices: G \subset 'C_G(A) by rewrite subsetI subxx centsC. rewrite -(quotientSGK nHG) ?subsetI ?sHG 1?centsC //. by rewrite coprim... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | stable_factor_cent | |
stable_series_cent gT (A G : {group gT}) s : last 1%G s :=: G -> (A.-stable).-series 1%G s -> coprime #|G| #|A| -> solvable G -> A \subset 'C(G). Proof. move=> <-{G}; elim/last_ind: s => /= [|s G IHs]; first by rewrite cents1. rewrite last_rcons rcons_path /= => /andP[/IHs{IHs}]. move: {s}(last _ _) => H IH_H nHGA coGA... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | stable_series_cent | |
coprime_nil_faithful_cent_stab gT (A G : {group gT}) : A \subset 'N(G) -> coprime #|G| #|A| -> nilpotent G -> let C := 'C_G(A) in 'C_G(C) \subset C -> A \subset 'C(G). Proof. move=> nGA coGA nilG C; rewrite subsetI subsetIl centsC /= -/C => cCA. pose N := 'N_G(C); have sNG: N \subset G by rewrite subsetIl. have sCG: C ... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_nil_faithful_cent_stab | |
coprime_odd_faithful_Ohm1 gT p (A G : {group gT}) : p.-group G -> A \subset 'N(G) -> coprime #|G| #|A| -> odd #|G| -> A \subset 'C('Ohm_1(G)) -> A \subset 'C(G). Proof. move=> pG nGA coGA oddG; rewrite !(centsC A) => cAG1. have [-> | ntG] := eqsVneq G 1; first exact: sub1G. have{oddG ntG} [p_pr oddp]: prime p /\ odd p.... | Theorem | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_odd_faithful_Ohm1 | |
coprime_odd_faithful_cent_abelem gT p (A G E : {group gT}) : E \in 'E_p(G) -> p.-group G -> A \subset 'N(G) -> coprime #|G| #|A| -> odd #|G| -> A \subset 'C('Ldiv_p('C_G(E))) -> A \subset 'C(G). Proof. case/pElemP=> sEG abelE pG nGA coGA oddG cCEA. have [-> | ntG] := eqsVneq G 1; first by rewrite cents1. have [p_pr _ _... | Corollary | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_odd_faithful_cent_abelem | |
critical_odd gT p (G : {group gT}) : p.-group G -> odd #|G| -> G :!=: 1 -> {H : {group gT} | [/\ H \char G, [~: H, G] \subset 'Z(H), nil_class H <= 2, exponent H = p & p.-group 'C(H | [Aut G])]}. Proof. move=> pG oddG ntG; have [H krH]:= Thompson_critical pG. have [chH sPhiZ sGH_Z scH] := krH; have clH := critical_clas... | Theorem | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | critical_odd | |
coprime_norm_quotient_pgroup : 'N(T / M) = 'N(T) / M. Proof. have [-> | ntT] := eqsVneq T 1; first by rewrite quotient1 !norm1 quotientT. have [p_pr _ [m oMpm]] := pgroup_pdiv pT ntT. apply/eqP; rewrite eqEsubset morphim_norms // andbT; apply/subsetP=> Mx. case: (cosetP Mx) => x Nx ->{Mx} nTqMx. have sylT: p.-Sylow(M <... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_norm_quotient_pgroup | |
coprime_cent_quotient_pgroup : 'C(T / M) = 'C(T) / M. Proof. symmetry; rewrite -quotientInorm -quotientMidl -['C(T / M)]cosetpreK. congr (_ / M); set Cq := _ @*^-1 _; set C := 'N_('C(T))(M). suffices <-: 'N_Cq(T) = C. rewrite setIC group_modl ?sub_cosetpre //= -/Cq; apply/setIidPr. rewrite -quotientSK ?subsetIl // cose... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_cent_quotient_pgroup | |
coprime_subnorm_quotient_pgroup : 'N_(G / M)(T / M) = 'N_G(T) / M. Proof. by rewrite quotientGI -?coprime_norm_quotient_pgroup. Qed. (* This is B & G, Lemma 1.14, for a local centraliser. *) | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_subnorm_quotient_pgroup | |
coprime_subcent_quotient_pgroup : 'C_(G / M)(T / M) = 'C_G(T) / M. Proof. by rewrite quotientGI -?coprime_cent_quotient_pgroup. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_subcent_quotient_pgroup | |
solvable_p_constrained : solvable G -> p.-constrained G. Proof. move=> solG P sylP; have [sPO pP _] := and3P sylP; pose K := 'O_p^'(G). have nKG: G \subset 'N(K) by rewrite normal_norm ?pcore_normal. have nKC: 'C_G(P) \subset 'N(K) by rewrite subIset ?nKG. rewrite -(quotientSGK nKC) //; last first. by rewrite /= -pseri... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | solvable_p_constrained | |
p_stable_abelian_constrained : p.-constrained G -> p.-stable G -> p.-abelian_constrained G. Proof. move=> constrG stabG P A sylP cAA /andP[sAP nAP]. have [sPG pP _] := and3P sylP; have sAG := subset_trans sAP sPG. set K2 := 'O_{p^', p}(G); pose K1 := 'O_p^'(G); pose Q := P :&: K2. have sQG: Q \subset G by rewrite subIs... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p_stable_abelian_constrained | |
p'core_cent_pgroup gT p (G R : {group gT}) : p.-subgroup(G) R -> solvable G -> 'O_p^'('C_G(R)) \subset 'O_p^'(G). Proof. case/andP=> sRG pR solG. without loss p'G1: gT G R sRG pR solG / 'O_p^'(G) = 1. have nOG_CR: 'C_G(R) \subset 'N('O_p^'(G)) by rewrite subIset ?gFnorm. move=> IH; rewrite -quotient_sub1 ?gFsub_trans /... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p'core_cent_pgroup | |
coprime_abelian_gen_cent gT (A G : {group gT}) : abelian A -> A \subset 'N(G) -> coprime #|G| #|A| -> <<\bigcup_(B : {group gT} | cyclic (A / B) && (B <| A)) 'C_G(B)>> = G. Proof. move=> abelA nGA coGA; symmetry; move: {2}_.+1 (ltnSn #|G|) => n. elim: n gT => // n IHn gT in A G abelA nGA coGA *; rewrite ltnS => leGn. w... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_abelian_gen_cent | |
coprime_abelian_gen_cent1 gT (A G : {group gT}) : abelian A -> ~~ cyclic A -> A \subset 'N(G) -> coprime #|G| #|A| -> <<\bigcup_(a in A^#) 'C_G[a]>> = G. Proof. move=> abelA ncycA nGA coGA. apply/eqP; rewrite eq_sym eqEsubset /= gen_subG. apply/andP; split; last by apply/bigcupsP=> B _; apply: subsetIl. rewrite -{1}(co... | Proposition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | coprime_abelian_gen_cent1 | |
focal_subgroup_gen : S :&: G^`(1) = <<[set [~ x, u] | x in S, u in G & x ^ u \in S]>>. Proof. set K := <<_>>; set G' := G^`(1); have [sSG coSiSG] := andP (pHall_Hall sylS). apply/eqP; rewrite eqEsubset gen_subG andbC; apply/andP; split. apply/subsetP=> _ /imset2P[x u Sx /setIdP[Gu Sxu] ->]. by rewrite inE groupM ?group... | Theorem | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | focal_subgroup_gen | |
Burnside_normal_complement : 'N_G(S) \subset 'C(S) -> 'O_p^'(G) ><| S = G. Proof. move=> cSN; set K := 'O_p^'(G); have [sSG pS _] := and3P sylS. have /andP[sKG nKG]: K <| G by apply: pcore_normal. have{nKG} nKS := subset_trans sSG nKG. have p'K: p^'.-group K by apply: pcore_pgroup. have{pS p'K} tiKS: K :&: S = 1 by rew... | Theorem | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Burnside_normal_complement | |
cyclic_Sylow_tiVsub_der1 : cyclic S -> S :&: G^`(1) = 1 \/ S \subset G^`(1). Proof. move=> cycS; have [sSG pS _] := and3P sylS. have nsSN: S <| 'N_G(S) by rewrite normalSG. have hallSN: Hall 'N_G(S) S. by apply: pHall_Hall (pHall_subl _ _ sylS); rewrite ?subsetIl ?normal_sub. have /splitsP[K /complP[tiSK /= defN]] := S... | Corollary | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | cyclic_Sylow_tiVsub_der1 | |
Zgroup_der1_Hall gT (G : {group gT}) : Zgroup G -> Hall G G^`(1). Proof. move=> ZgG; set G' := G^`(1). rewrite /Hall der_sub coprime_sym coprime_pi' ?cardG_gt0 //=. apply/pgroupP=> p p_pr pG'; have [P sylP] := Sylow_exists p G. have cycP: cyclic P by have:= forallP ZgG P; rewrite (p_Sylow sylP). case: (cyclic_Sylow_tiV... | Corollary | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Zgroup_der1_Hall | |
cyclic_pdiv_normal_complement gT (S G : {group gT}) : (pdiv #|G|).-Sylow(G) S -> cyclic S -> exists H : {group gT}, H ><| S = G. Proof. set p := pdiv _ => sylS cycS; have cSS := cyclic_abelian cycS. exists 'O_p^'(G)%G; apply: Burnside_normal_complement => //. have [-> | ntS] := eqsVneq S 1; first apply: cents1. have [s... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | cyclic_pdiv_normal_complement | |
Zgroup_metacyclic gT (G : {group gT}) : Zgroup G -> metacyclic G. Proof. elim: {G}_.+1 {-2}G (ltnSn #|G|) => // n IHn G; rewrite ltnS => leGn ZgG. have{n IHn leGn} solG: solvable G. have [-> | ntG] := eqsVneq G 1; first apply: solvable1. have [S sylS] := Sylow_exists (pdiv #|G|) G. have cycS: cyclic S := forall_inP ZgG... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Zgroup_metacyclic | |
Maschke_abelem gT p (G V U : {group gT}) : p.-abelem V -> p^'.-group G -> U \subset V -> G \subset 'N(V) -> G \subset 'N(U) -> exists2 W : {group gT}, U \x W = V & G \subset 'N(W). Proof. move=> pV p'G sUV nVG nUG. have splitU: [splits V, over U] := abelem_splits pV sUV. case/and3P: pV => pV abV; have cUV := subset_tra... | Theorem | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Maschke_abelem | |
plength1_1 : p.-length_1 (1 : {set gT}). Proof. by rewrite -[_ 1]subG1 pseries_sub. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | plength1_1 | |
plength1_p'group G : p^'.-group G -> p.-length_1 G. Proof. move=> p'G; rewrite [p.-length_1 G]eqEsubset pseries_sub /=. by rewrite -{1}(pcore_pgroup_id p'G) -pseries1 pseries_sub_catl. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | plength1_p'group | |
plength1_nonprime G : ~~ prime p -> p.-length_1 G. Proof. move=> not_p_pr; rewrite plength1_p'group // p'groupEpi mem_primes. by rewrite (negPf not_p_pr). Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | plength1_nonprime | |
plength1_pcore_quo_Sylow G (Gb := G / 'O_p^'(G)) : p.-length_1 G = p.-Sylow(Gb) 'O_p(Gb). Proof. rewrite /plength_1 eqEsubset pseries_sub /=. rewrite (pseries_rcons _ [:: _; _]) -sub_quotient_pre ?gFnorm //=. rewrite /pHall pcore_sub pcore_pgroup /= -card_quotient ?gFnorm //=. rewrite -quotient_pseries2 /= {}/Gb -(pser... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | plength1_pcore_quo_Sylow | |
plength1_pcore_Sylow G : 'O_p^'(G) = 1 -> p.-length_1 G = p.-Sylow(G) 'O_p(G). Proof. move=> p'G1; rewrite plength1_pcore_quo_Sylow -quotient_pseries2. by rewrite p'G1 pseries_pop2 // pquotient_pHall ?normal1 ?pgroup1. Qed. (* This is the characterization given in Section 10 of B & G, p. 75, just *) (* before Theorem 1... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | plength1_pcore_Sylow | |
plength1_pseries2_quo G : p.-length_1 G = p^'.-group (G / 'O_{p^', p}(G)). Proof. rewrite /plength_1 eqEsubset pseries_sub lastI pseries_rcons /=. rewrite -sub_quotient_pre ?gFnorm //. by apply/idP/idP=> pl1G; rewrite ?pcore_pgroup_id ?(pgroupS pl1G) ?pcore_pgroup. Qed. (* This is B & G, Lemma 1.21(a). *) | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | plength1_pseries2_quo | |
plength1S G H : H \subset G -> p.-length_1 G -> p.-length_1 H. Proof. rewrite /plength_1 => sHG pG1; rewrite eqEsubset pseries_sub. by apply: subset_trans (pseriesS _ sHG); rewrite (eqP pG1) (setIidPr _). Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | plength1S | |
plength1_quo G H : p.-length_1 G -> p.-length_1 (G / H). Proof. rewrite /plength_1 => pG1; rewrite eqEsubset pseries_sub. by rewrite -{1}(eqP pG1) morphim_pseries. Qed. (* This is B & G, Lemma 1.21(b). *) | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | plength1_quo | |
p'quo_plength1 G H : H <| G -> p^'.-group H -> p.-length_1 (G / H) = p.-length_1 G. Proof. rewrite /plength_1 => nHG p'H; apply/idP/idP; last exact: plength1_quo. move=> pGH1; rewrite eqEsubset pseries_sub. have nOG: 'O_{p^'}(G) <| G by apply: pseries_normal. rewrite -(quotientSGK (normal_norm nOG)) ?(pseries_sub_catl ... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p'quo_plength1 | |
pquo_plength1 G H : H <| G -> p.-group H -> 'O_p^'(G / H) = 1-> p.-length_1 (G / H) = p.-length_1 G. Proof. rewrite /plength_1 => nHG pH trO; apply/idP/idP; last exact: plength1_quo. rewrite (pseries_pop _ trO) => pGH1; rewrite eqEsubset pseries_sub /=. rewrite pseries_pop //; last first. apply/eqP; rewrite -subG1; hav... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | pquo_plength1 | |
p_elt_gen_group A : {group gT} := Eval hnf in [group of p_elt_gen p A]. (* Note that p_elt_gen could be a functor. *) | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p_elt_gen_group | |
p_elt_gen_normal G : p_elt_gen p G <| G. Proof. apply/normalP; split=> [|x Gx]. by rewrite gen_subG; apply/subsetP=> x; rewrite inE; case/andP. rewrite -genJ; congr <<_>>; apply/setP=> y; rewrite mem_conjg !inE. by rewrite p_eltJ -mem_conjg conjGid. Qed. (* This is B & G, Lemma 1.21(d). *) | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p_elt_gen_normal | |
p_elt_gen_length1 G : p.-length_1 G = p^'.-Hall(p_elt_gen p G) 'O_p^'(p_elt_gen p G). Proof. rewrite /pHall pcore_sub pcore_pgroup pnatNK /= /plength_1. have nUG := p_elt_gen_normal G; have [sUG nnUG]:= andP nUG. apply/idP/idP=> [p1G | pU]. apply: (@pnat_dvd _ #|p_elt_gen p G : 'O_p^'(G)|). by rewrite -[#|_ : 'O_p^'(G)... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | p_elt_gen_length1 | |
quo2_plength1 gT p (G H K : {group gT}) : H <| G -> K <| G -> H :&: K = 1 -> p.-length_1 (G / H) && p.-length_1 (G / K) = p.-length_1 G. Proof. move=> nHG nKG trHK. have [p_pr | p_nonpr] := boolP (prime p); last by rewrite !plength1_nonprime. apply/andP/idP=> [[pH1 pK1] | pG1]; last by rewrite !plength1_quo. pose U := ... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | quo2_plength1 | |
logn_quotient_cent_abelem gT p (A E : {group gT}) : A \subset 'N(E) -> p.-abelem E -> logn p #|E| <= 2 -> logn p #|A : 'C_A(E)| <= 1. Proof. move=> nEA abelE maxdimE; have [-> | ntE] := eqsVneq E 1. by rewrite (setIidPl (cents1 _)) indexgg logn1. pose rP := abelem_repr abelE ntE nEA. have [p_pr _ _] := pgroup_pdiv (abe... | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | logn_quotient_cent_abelem | |
Puig_succ_group gT (D E : {set gT}) := [group of 'L_[D](E)]. | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_succ_group | |
Puig_at_group_set n gT D : @group_set gT 'L_{n}(D). Proof. by case: n => [|n]; apply: groupP. Qed. | Fact | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_at_group_set | |
Puig_at_group n gT D := Group (@Puig_at_group_set n gT D). | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_at_group | |
Puig_inf_group gT (D : {set gT}) := [group of 'L_*(D)]. | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_inf_group | |
Puig_group gT (D : {set gT}) := [group of 'L(D)]. | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_group | |
Puig0 D : 'L_{0}(D) = 1. Proof. by []. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig0 | |
PuigS n D : 'L_{n.+1}(D) = 'L_[D]('L_{n}(D)). Proof. by []. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | PuigS | |
Puig_recE n D : Puig_rec n D = 'L_{n}(D). Proof. by []. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_recE | |
Puig_def D : 'L(D) = 'L_[D]('L_*(D)). Proof. by []. Qed. Local Notation "D --> E" := (generated_by (norm_abelian D) E) (at level 70, no associativity) : group_scope. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_def | |
Puig_gen D E : E --> 'L_[D](E). Proof. by apply/existsP; exists (subgroups D). Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_gen | |
Puig_max G D E : D --> E -> E \subset G -> E \subset 'L_[G](D). Proof. case/existsP=> gE /eqP <-{E}; rewrite !gen_subG. move/bigcupsP=> sEG; apply/bigcupsP=> A gEA; have [_ abnA]:= andP gEA. by rewrite sub_gen // bigcup_sup // inE sEG. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_max | |
norm_abgenS D1 D2 E : D1 \subset D2 -> D2 --> E -> D1 --> E. Proof. move=> sD12 /exists_eqP[gE <-{E}]. apply/exists_eqP; exists [set A in gE | norm_abelian D2 A]. congr <<_>>; apply: eq_bigl => A; rewrite !inE. apply: andb_idr => /and3P[_ nAD cAA]. by apply/andP; rewrite (subset_trans sD12). Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | norm_abgenS | |
Puig_succ_sub G D : 'L_[G](D) \subset G. Proof. by rewrite gen_subG; apply/bigcupsP=> A /andP[]; rewrite inE. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_succ_sub | |
Puig_at_sub n G : 'L_{n}(G) \subset G. Proof. by case: n => [|n]; rewrite ?sub1G ?Puig_succ_sub. Qed. (* This is B & G, Lemma B.1(d), first part. *) | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_at_sub | |
Puig_inf_sub G : 'L_*(G) \subset G. Proof. exact: Puig_at_sub. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_inf_sub | |
Puig_sub G : 'L(G) \subset G. Proof. exact: Puig_at_sub. Qed. (* This is part of B & G, Lemma B.1(b). *) | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_sub | |
Puig1 G : 'L_{1}(G) = G. Proof. apply/eqP; rewrite eqEsubset Puig_at_sub; apply/subsetP=> x Gx. rewrite -cycle_subG sub_gen // -[<[x]>]/(gval _) bigcup_sup //=. by rewrite inE cycle_subG Gx /= /norm_abelian cycle_abelian sub1G. Qed. | Lemma | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig1 | |
Puig_at_cont n : GFunctor.iso_continuous (Puig_at n). Proof. elim: n => [|n IHn] aT rT G f injf; first by rewrite morphim1. have IHnS := Puig_at_sub n; pose func_n := [igFun by IHnS & !IHn]. rewrite !PuigS sub_morphim_pre ?Puig_succ_sub // gen_subG; apply/bigcupsP=> A. rewrite inE => /and3P[sAG nAL cAA]; rewrite -sub_m... | Fact | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_at_cont | |
Puig_at_igFun n := [igFun by Puig_at_sub^~ n & !Puig_at_cont n]. | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_at_igFun | |
Puig_inf_cont : GFunctor.iso_continuous Puig_inf. Proof. by move=> aT rT G f injf; rewrite /Puig_inf card_injm // Puig_at_cont. Qed. | Fact | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_inf_cont | |
Puig_inf_igFun := [igFun by Puig_inf_sub & !Puig_inf_cont]. | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_inf_igFun | |
Puig_cont : GFunctor.iso_continuous Puig. Proof. by move=> aT rT G f injf; rewrite /Puig card_injm // Puig_at_cont. Qed. | Fact | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_cont | |
Puig_igFun := [igFun by Puig_sub & !Puig_cont]. | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset binomial.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.",
... | theories/BGsection1.v | Puig_igFun | |
alpha := [pred p | 2 < 'r_p(M)]. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct.",
"From mathcomp Require Import cyclic... | theories/BGsection10.v | alpha | |
alpha_core := 'O_alpha(M). | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct.",
"From mathcomp Require Import cyclic... | theories/BGsection10.v | alpha_core | |
Structure alpha_core_group := Eval hnf in [group of alpha_core]. | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct.",
"From mathcomp Require Import cyclic... | theories/BGsection10.v | Structure | |
beta := [pred p | [forall (P : {group gT} | p.-Sylow(M) P), ~~ p.-narrow P]]. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct.",
"From mathcomp Require Import cyclic... | theories/BGsection10.v | beta | |
beta_core := 'O_beta(M). | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct.",
"From mathcomp Require Import cyclic... | theories/BGsection10.v | beta_core | |
Structure beta_core_group := Eval hnf in [group of beta_core]. | Canonical | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct.",
"From mathcomp Require Import cyclic... | theories/BGsection10.v | Structure | |
sigma := [pred p | [exists (P : {group gT} | p.-Sylow(M) P), 'N(P) \subset M]]. | Definition | theories | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.",
"From mathcomp Require Import div fintype bigop prime finset.",
"From mathcomp Require Import fingroup morphism perm automorphism quotient.",
"From mathcomp Require Import action gproduct.",
"From mathcomp Require Import cyclic... | theories/BGsection10.v | sigma |
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