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 |
|---|---|---|---|---|---|---|
sigma_core := 'O_sigma(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_core | |
Structure sigma_core_group := Eval hnf in [group of sigma_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 | |
sigmaJ x : \sigma(H :^ x) =i \sigma(H). Proof. move=> p; apply/exists_inP/exists_inP=> [] [P sylP sNH]; last first. by exists (P :^ x)%G; rewrite ?pHallJ2 // normJ conjSg. by exists (P :^ x^-1)%G; rewrite ?normJ ?sub_conjgV // -(pHallJ2 _ _ _ x) actKV. Qed. | Lemma | 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 | sigmaJ | |
MsigmaJ x : (H :^ x)`_\sigma = H`_\sigma :^ x. Proof. by rewrite /sigma_core -(eq_pcore H (sigmaJ x)) pcoreJ. Qed. | Lemma | 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 | MsigmaJ | |
alphaJ x : \alpha(H :^ x) =i \alpha(H). Proof. by move=> p; rewrite !inE /= p_rankJ. Qed. | Lemma | 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 | alphaJ | |
MalphaJ x : (H :^ x)`_\alpha = H`_\alpha :^ x. Proof. by rewrite /alpha_core -(eq_pcore H (alphaJ x)) pcoreJ. Qed. | Lemma | 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 | MalphaJ | |
betaJ x : \beta(H :^ x) =i \beta(H). Proof. move=> p; apply/forall_inP/forall_inP=> nnSylH P sylP. by rewrite -(@narrowJ _ _ _ x) nnSylH ?pHallJ2. by rewrite -(@narrowJ _ _ _ x^-1) nnSylH // -(pHallJ2 _ _ _ x) actKV. Qed. | Lemma | 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 | betaJ | |
MbetaJ x : (H :^ x)`_\beta = H`_\beta :^ x. Proof. by rewrite /beta_core -(eq_pcore H (betaJ x)) pcoreJ. Qed. | Lemma | 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 | MbetaJ | |
not_narrow_ideal p P : p.-Sylow(G) P -> ~~ p.-narrow P -> p \in \beta(G). Proof. move=> sylP nnP; apply/forall_inP=> Q sylQ. by have [x _ ->] := Sylow_trans sylP sylQ; rewrite narrowJ. Qed. | Remark | 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 | not_narrow_ideal | |
beta_sub_alpha : {subset \beta(M) <= \alpha(M)}. Proof. move=> p; rewrite !inE /= => /forall_inP nnSylM. have [P sylP] := Sylow_exists p M; have:= nnSylM P sylP. by rewrite negb_imply (p_rank_Sylow sylP) => /andP[]. Qed. | Remark | 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_sub_alpha | |
alpha_sub_sigma : {subset \alpha(M) <= \sigma(M)}. Proof. move=> p a_p; have [P sylP] := Sylow_exists p M; have [sPM pP _ ] := and3P sylP. have{a_p} rP: 2 < 'r(P) by rewrite (rank_Sylow sylP). apply/exists_inP; exists P; rewrite ?uniq_mmax_norm_sub //. exact: def_uniq_mmax (rank3_Uniqueness (mFT_pgroup_proper pP) rP) m... | Remark | 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_sub_sigma | |
beta_sub_sigma : {subset \beta(M) <= \sigma(M)}. Proof. by move=> p; move/beta_sub_alpha; apply: alpha_sub_sigma. Qed. | Remark | 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_sub_sigma | |
Mbeta_sub_Malpha : M`_\beta \subset M`_\alpha. Proof. exact: sub_pcore beta_sub_alpha. Qed. | Remark | 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 | Mbeta_sub_Malpha | |
Malpha_sub_Msigma : M`_\alpha \subset M`_\sigma. Proof. exact: sub_pcore alpha_sub_sigma. Qed. | Remark | 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 | Malpha_sub_Msigma | |
Mbeta_sub_Msigma : M`_\beta \subset M`_\sigma. Proof. exact: sub_pcore beta_sub_sigma. Qed. (* This is the first part of the remark just above B & G, Theorem 10.1. *) | Remark | 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 | Mbeta_sub_Msigma | |
norm_sigma_Sylow p P : p \in \sigma(M) -> p.-Sylow(M) P -> 'N(P) \subset M. Proof. case/exists_inP=> Q sylQ sNPM sylP. by case: (Sylow_trans sylQ sylP) => m mM ->; rewrite normJ conj_subG. Qed. (* This is the second part of the remark just above B & G, Theorem 10.1. *) | Remark | 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 | norm_sigma_Sylow | |
sigma_Sylow_G p P : p \in \sigma(M) -> p.-Sylow(M) P -> p.-Sylow(G) P. Proof. move=> sMp sylP; apply: (mmax_sigma_Sylow maxM) => //. exact: norm_sigma_Sylow sMp sylP. Qed. | Remark | 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_Sylow_G | |
sigma_Sylow_neq1 p P : p \in \sigma(M) -> p.-Sylow(M) P -> P :!=: 1. Proof. move=> sMp /(norm_sigma_Sylow sMp); apply: contraTneq => ->. by rewrite norm1 subTset -properT mmax_proper. Qed. | Lemma | 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_Sylow_neq1 | |
sigma_sub_pi : {subset \sigma(M) <= \pi(M)}. Proof. move=> p sMp; have [P sylP]:= Sylow_exists p M. by rewrite -p_rank_gt0 -(rank_Sylow sylP) rank_gt0 (sigma_Sylow_neq1 sMp sylP). Qed. | Lemma | 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_sub_pi | |
predI_sigma_alpha : [predI \sigma(M) & \alpha(G)] =i \alpha(M). Proof. move=> p; rewrite inE /= -(andb_idl (@alpha_sub_sigma p)). apply: andb_id2l => sMp; have [P sylP] := Sylow_exists p M. by rewrite !inE -(rank_Sylow sylP) -(rank_Sylow (sigma_Sylow_G sMp sylP)). Qed. | Lemma | 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 | predI_sigma_alpha | |
predI_sigma_beta : [predI \sigma(M) & \beta(G)] =i \beta(M). Proof. move=> p; rewrite inE /= -(andb_idl (@beta_sub_sigma p)). apply: andb_id2l => sMp; apply/idP/forall_inP=> [bGp P sylP | nnSylM]. exact: forall_inP bGp P (sigma_Sylow_G sMp sylP). have [P sylP] := Sylow_exists p M. exact: not_narrow_ideal (sigma_Sylow_G... | Lemma | 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 | predI_sigma_beta | |
sigma_Sylow_trans M p X g : p \in \sigma(M) -> p.-Sylow(M) X -> X :^ g \subset M -> g \in M. Proof. move=> sMp sylX sXgM; have pX := pHall_pgroup sylX. have [|h hM /= sXghX] := Sylow_Jsub sylX sXgM; first by rewrite pgroupJ. by rewrite -(groupMr _ hM) (subsetP (norm_sigma_Sylow _ sylX)) ?inE ?conjsgM. Qed. (* This is B... | Theorem | 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_Sylow_trans | |
sigma_group_trans M p X : M \in 'M -> p \in \sigma(M) -> p.-group X -> [/\ (*a*) forall g, X \subset M -> X :^ g \subset M -> exists2 c, c \in 'C(X) & exists2 m, m \in M & g = c * m, (*b*) [transitive 'C(X), on [set Mg in M :^: G | X \subset Mg] | 'Js ] & (*c*) X \subset M -> 'C(X) * 'N_M(X) = 'N(X)]. Proof. move=> max... | Theorem | 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_group_trans | |
ltMG := mmax_proper maxM. | Let | 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 | ltMG | |
solM := mmax_sol maxM. | Let | 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 | solM | |
aMa : \alpha(M).-group (M`_\alpha). Proof. exact: pcore_pgroup. Qed. | Let | 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 | aMa | |
nsMaM : M`_\alpha <| M. Proof. exact: pcore_normal. Qed. | Let | 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 | nsMaM | |
sMaMs : M`_\alpha \subset M`_\sigma. Proof. exact: Malpha_sub_Msigma. Qed. | Let | 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 | sMaMs | |
F := 'F(M / M`_\alpha). | Let | 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 | F | |
nsFMa : F <| M / M`_\alpha. Proof. exact: Fitting_normal. Qed. | Let | 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 | nsFMa | |
alpha'F : \alpha(M)^'.-group F. Proof. rewrite -[F](nilpotent_pcoreC \alpha(M) (Fitting_nil _)) -Fitting_pcore /=. by rewrite trivg_pcore_quotient (trivgP (Fitting_sub 1)) dprod1g pcore_pgroup. Qed. | Let | 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'F | |
Malpha_quo_sub_Fitting : M^`(1) / M`_\alpha \subset F. Proof. have [/= K defF sMaK nsKM] := inv_quotientN nsMaM nsFMa; rewrite -/F in defF. have [sKM _] := andP nsKM; have nsMaK: M`_\alpha <| K := normalS sMaK sKM nsMaM. have [[_ nMaK] [_ nMaM]] := (andP nsMaK, andP nsMaM). have hallMa: \alpha(M).-Hall(K) M`_\alpha. by... | Let | 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 | Malpha_quo_sub_Fitting | |
sigma_Hall_sub_der1 H : \sigma(M).-Hall(M) H -> H \subset M^`(1). Proof. move=> hallH; have [sHM sH _] := and3P hallH. rewrite -(Sylow_gen H) gen_subG; apply/bigcupsP=> P /SylowP[p p_pr sylP]. have [-> | ntP] := eqsVneq P 1; first by rewrite sub1G. have [sPH pP _] := and3P sylP; have{ntP} [_ p_dv_P _] := pgroup_pdiv pP... | Let | 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_Hall_sub_der1 | |
Malpha_Hall : \alpha(M).-Hall(M) M`_\alpha. Proof. have [H hallH] := Hall_exists \sigma(M) solM; have [sHM sH _] := and3P hallH. rewrite (subHall_Hall hallH (alpha_sub_sigma maxM)) // /pHall pcore_pgroup /=. rewrite -(card_quotient (subset_trans sHM (normal_norm nsMaM))) -pgroupE. rewrite (subset_trans sMaMs) ?pcore_su... | Theorem | 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 | Malpha_Hall | |
Msigma_Hall : \sigma(M).-Hall(M) M`_\sigma. Proof. have [H hallH] := Hall_exists \sigma(M) solM; have [sHM sH _] := and3P hallH. rewrite /M`_\sigma (normal_Hall_pcore hallH) // -(quotientGK nsMaM). rewrite -(quotientGK (normalS _ sHM nsMaM)) ?cosetpre_normal //; last first. by rewrite (subset_trans sMaMs) ?pcore_sub_Ha... | Theorem | 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 | Msigma_Hall | |
pi_Msigma : \pi(M`_\sigma) =i \sigma(M). Proof. move=> p; apply/idP/idP=> [|s_p /=]; first exact: pnatPpi (pcore_pgroup _ _). by rewrite (card_Hall Msigma_Hall) pi_of_part // inE /= sigma_sub_pi. Qed. (* This is B & G, Theorem 10.2(b2). *) | Lemma | 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 | pi_Msigma | |
Msigma_Hall_G : \sigma(M).-Hall(G) M`_\sigma. Proof. rewrite pHallE subsetT /= eqn_dvd {1}(card_Hall Msigma_Hall). rewrite partn_dvd ?cardG_gt0 ?cardSg ?subsetT //=. apply/dvdn_partP; rewrite ?part_gt0 // => p. rewrite pi_of_part ?cardG_gt0 // => /andP[_ s_p]. rewrite partn_part => [|q /eqnP-> //]. have [P sylP] := Syl... | Theorem | 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 | Msigma_Hall_G | |
Malpha_Hall_G : \alpha(M).-Hall(G) M`_\alpha. Proof. apply: subHall_Hall Msigma_Hall_G (alpha_sub_sigma maxM) _. exact: pHall_subl sMaMs (pcore_sub _ _) Malpha_Hall. Qed. (* This is B & G, Theorem 10.2(c). *) | Theorem | 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 | Malpha_Hall_G | |
Msigma_der1 : M`_\sigma \subset M^`(1). Proof. exact: sigma_Hall_sub_der1 Msigma_Hall. Qed. (* This is B & G, Theorem 10.2(d1). *) | Theorem | 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 | Msigma_der1 | |
Malpha_quo_rank2 : 'r(M / M`_\alpha) <= 2. Proof. have [p p_pr ->] := rank_witness (M / M`_\alpha). have [P sylP] := Sylow_exists p M; have [sPM pP _] := and3P sylP. have nMaP := subset_trans sPM (normal_norm nsMaM). rewrite -(rank_Sylow (quotient_pHall nMaP sylP)) /= leqNgt. have [a_p | a'p] := boolP (p \in \alpha(M))... | Theorem | 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 | Malpha_quo_rank2 | |
Malpha_quo_nil : nilpotent (M^`(1) / M`_\alpha). Proof. exact: nilpotentS Malpha_quo_sub_Fitting (Fitting_nil _). Qed. (* This is B & G, Theorem 10.2(e). *) | Theorem | 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 | Malpha_quo_nil | |
Msigma_neq1 : M`_\sigma :!=: 1. Proof. without loss Ma1: / M`_\alpha = 1. by case: eqP => // Ms1 -> //; apply/trivgP; rewrite -Ms1 Malpha_sub_Msigma. have{Ma1} rFM: 'r('F(M)) <= 2. rewrite (leq_trans _ Malpha_quo_rank2) // Ma1. by rewrite -(isog_rank (quotient1_isog _)) rankS ?Fitting_sub. pose q := max_pdiv #|M|; pose... | Theorem | 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 | Msigma_neq1 | |
cent_alpha'_uniq X : X \subset M -> \alpha(M)^'.-group X -> 'r('C_(M`_\alpha)(X)) >= 2 -> 'C_M(X)%G \in 'U. Proof. have ltM_G := sub_proper_trans (subsetIl M _) ltMG. move=> sXM a'X; have [p p_pr -> rCX] := rank_witness 'C_(M`_\alpha)(X). have{rCX} [B EpB] := p_rank_geP rCX; have{EpB} [sBCX abelB dimB] := pnElemP EpB. ... | Theorem | 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 | cent_alpha'_uniq | |
der1_quo_sigma' : p %| #|M / M^`(1)| -> p \in \sigma(M)^'. Proof. apply: contraL => /= s_p; have piMp := sigma_sub_pi maxM s_p. have p_pr: prime p by move: piMp; rewrite mem_primes; case/andP. rewrite -p'natE ?(pi'_p'nat _ s_p) // -pgroupE -partG_eq1. rewrite -(card_Hall (quotient_pHall _ Msigma_Hall)) /=; last first. ... | Lemma | 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 | der1_quo_sigma' | |
cent1_sigma'_Zgroup P : p.-Sylow(M) P -> P :!=: 1 -> exists x, [/\ x \in 'Ohm_1('Z(P))^#, 'M('C[x]) != [set M] & Zgroup 'C_(M`_\alpha)[x]]. Proof. move=> sylP ntP; have [sPM pP _] := and3P sylP; have nilP := pgroup_nil pP. set T := 'Ohm_1('Z(P)); have charT: T \char P by rewrite !gFchar_trans. suffices [x Tx not_uCx]: ... | Lemma | 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 | cent1_sigma'_Zgroup | |
sigma'_rank2_max : 'r_p(M) = 2 -> 'E_p^2(M) \subset 'E*_p(G). Proof. move=> rpM; apply: contraR s'p => /subsetPn[A Ep2A not_maxA]. have{Ep2A} [sAM abelA dimA] := pnElemP Ep2A; have [pA _ _] := and3P abelA. have [P sylP sAP] := Sylow_superset sAM pA; have [_ pP _] := and3P sylP. apply/exists_inP; exists P; rewrite ?uniq... | Lemma | 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'_rank2_max | |
sigma'_rank2_beta' : 'r_p(M) = 2 -> p \notin \beta(G). Proof. move=> rpM; rewrite -[p \in _]negb_exists_in negbK; apply/exists_inP. have [A Ep2A]: exists A, A \in 'E_p^2(M) by apply/p_rank_geP; rewrite rpM. have [_ abelA dimA] := pnElemP Ep2A; have [pA _] := andP abelA. have [P sylP sAP] := Sylow_superset (subsetT _) p... | Lemma | 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'_rank2_beta' | |
sigma'_norm_mmax_rank2 X : p.-group X -> 'N(X) \subset M -> 'r_p(M) = 2. Proof. move=> pX sNX_M; have sXM: X \subset M := subset_trans (normG X) sNX_M. have [P sylP sXP] := Sylow_superset sXM pX; have [sPM pP _] := and3P sylP. apply: contraNeq s'p; case: ltngtP => // rM _; last exact: alpha_sub_sigma. apply/exists_inP;... | Lemma | 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'_norm_mmax_rank2 | |
sigma'1Elem_sub_p2Elem X : X \in 'E_p^1(G) -> 'N(X) \subset M -> exists2 A, A \in 'E_p^2(G) & X \subset A. Proof. move=> EpX sNXM; have sXM := subset_trans (normG X) sNXM. have [[_ abelX dimX] p_pr] := (pnElemP EpX, pnElem_prime EpX). have pX := abelem_pgroup abelX; have rpM2 := sigma'_norm_mmax_rank2 pX sNXM. have [P ... | Lemma | 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'1Elem_sub_p2Elem | |
mFT_proper_plength1 p H : H \proper G -> p.-length_1 H. Proof. case/mmax_exists=> M /setIdP[maxM sHM]. suffices{H sHM}: p.-length_1 M by apply: plength1S. have [solM oddM] := (mmax_sol maxM, mFT_odd M). have [rpMle2 | a_p] := leqP 'r_p(M) 2. by rewrite plength1_pseries2_quo; case/rank2_der1_complement: rpMle2. pose Ma ... | Theorem | 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 | mFT_proper_plength1 | |
pP : p.-group P := pHall_pgroup sylP_G. (* This is an B & G, Corollary 10.7(a), second part (which does not depend on *) (* a particular complement). *) | Let | 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 | pP | |
mFT_Sylow_der1 : P \subset 'N(P)^`(1). Proof. have [-> | ntP] := eqsVneq P 1; first exact: sub1G. have ltNG: 'N(P) \proper G := mFT_norm_proper ntP (mFT_pgroup_proper pP). have [M /setIdP[/= maxM sNM]] := mmax_exists ltNG. have [ltMG solM] := (mmax_proper maxM, mmax_sol maxM). have [pl1M sPM] := (mFT_proper_plength1 p ... | Corollary | 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 | mFT_Sylow_der1 | |
mFT_Sylow_sdprod_commg V : P ><| V = 'N(P) -> [~: P, V] = P. Proof. move=> defV; have sPN' := mFT_Sylow_der1. have sylP := pHall_subl (normG P) (subsetT 'N(P)) sylP_G. have [|//] := coprime_der1_sdprod defV _ (pgroup_sol pP) sPN'. by rewrite (coprime_sdprod_Hall_l defV) // (pHall_Hall sylP). Qed. (* This is B & G, Coro... | Corollary | 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 | mFT_Sylow_sdprod_commg | |
mFT_rank2_Sylow_cprod : 'r(P) < 3 -> ~~ abelian P -> exists2 S, [/\ ~~ abelian (gval S), logn p #|S| = 3 & exponent S %| p] & exists2 C, cyclic (gval C) & S \* C = P /\ 'Ohm_1(C) = 'Z(S). Proof. move=> rP not_cPP; have sylP := pHall_subl (normG P) (subsetT 'N(P)) sylP_G. have ntP: P :!=: 1 by apply: contraNneq not_cPP ... | Corollary | 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 | mFT_rank2_Sylow_cprod | |
mFT_sub_Sylow_trans : forall Q x, Q \subset P -> Q :^ x \subset P -> exists2 y, y \in 'N(P) & Q :^ x = Q :^ y. Proof. move=> Q x; have [-> /trivgP-> /trivgP-> | ntP sQP sQxP] := eqsVneq P 1. by exists 1; rewrite ?group1 ?conjs1g. have ltNG: 'N(P) \proper G := mFT_norm_proper ntP (mFT_pgroup_proper pP). have [M /=] := m... | Corollary | 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 | mFT_sub_Sylow_trans | |
mFT_subnorm_Sylow Q : Q \subset P -> p.-Sylow('N(Q)) 'N_P(Q). Proof. move=> sQP; have pQ := pgroupS sQP pP. have [S /= sylS] := Sylow_exists p 'N(Q); have [sNS pS _] := and3P sylS. have sQS: Q \subset S := normal_sub_max_pgroup (Hall_max sylS) pQ (normalG Q). have [x _ sSxP] := Sylow_Jsub sylP_G (subsetT S) pS. have sQ... | Corollary | 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 | mFT_subnorm_Sylow | |
mFT_Sylow_normalS Q R : p.-group R -> Q \subset P :&: R -> Q <| 'N(P) -> Q <| 'N(R). Proof. move=> pR /subsetIP[sQP sQR] /andP[nQP nQ_NP]. have [x _ sRxP] := Sylow_Jsub sylP_G (subsetT R) pR. rewrite /normal normsG //; apply/subsetP=> y nRy. have sQxP: Q :^ x \subset P by rewrite (subset_trans _ sRxP) ?conjSg. have sQy... | Corollary | 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 | mFT_Sylow_normalS | |
solM : solvable M := mmax_sol maxM. | Let | 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 | solM | |
ltMG : M \proper G := mmax_proper maxM. | Let | 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 | ltMG | |
sMbMs : M`_\beta \subset M`_\sigma := Mbeta_sub_Msigma maxM. | Let | 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 | sMbMs | |
nsMbM : M`_\beta <| M := pcore_normal _ _. | Let | 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 | nsMbM | |
hallMs : \sigma(M).-Hall(M) M`_\sigma := Msigma_Hall maxM. | Let | 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 | hallMs | |
nsMsM : M`_\sigma <| M := pcore_normal _ M. | Let | 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 | nsMsM | |
sMsM' : M`_\sigma \subset M^`(1) := Msigma_der1 maxM. | Let | 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 | sMsM' | |
Mbeta_der1 : M`_\beta \subset M^`(1). Proof. exact: subset_trans sMbMs sMsM'. Qed. | Lemma | 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 | Mbeta_der1 | |
sM'M : M^`(1) \subset M := der_sub 1 M. | Let | 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 | sM'M | |
nsMsM' : M`_\sigma <| M^`(1) := normalS sMsM' sM'M nsMsM. | Let | 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 | nsMsM' | |
nsMbM' : M`_\beta <| M^`(1) := normalS Mbeta_der1 sM'M nsMbM. | Let | 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 | nsMbM' | |
nMbM' := normal_norm nsMbM'. (* This is B & G, Lemma 10.8(c). *) | Let | 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 | nMbM' | |
beta_max_pdiv p : p \notin \beta(M) -> [/\ p^'.-Hall(M^`(1)) 'O_p^'(M^`(1)), p^'.-Hall(M`_\sigma) 'O_p^'(M`_\sigma) & forall q, q \in \pi(M / 'O_p^'(M)) -> q <= p]. Proof. rewrite !inE -negb_exists_in negbK => /exists_inP[P sylP nnP]. have [|ncM' p_max] := narrow_der1_complement_max_pdiv (mFT_odd M) solM sylP nnP. by r... | Lemma | 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_max_pdiv | |
Mbeta_Hall : \beta(M).-Hall(M) M`_\beta. Proof. have [H hallH] := Hall_exists \beta(M) solM; have [sHM bH _]:= and3P hallH. rewrite [M`_\beta](sub_pHall hallH) ?pcore_pgroup ?pcore_sub //=. rewrite -(setIidPl sMbMs) pcore_setI_normal ?pcore_normal //. have sH: \sigma(M).-group H := sub_pgroup (beta_sub_sigma maxM) bH. ... | Lemma | 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 | Mbeta_Hall | |
Mbeta_Hall_G : \beta(M).-Hall(G) M`_\beta. Proof. apply: (subHall_Hall (Msigma_Hall_G maxM) (beta_sub_sigma maxM)). exact: pHall_subl sMbMs (pcore_sub _ _) Mbeta_Hall. Qed. (* This is an equivalent form of B & G, Lemma 10.8(b), which is used directly *) (* later in the proof (e.g., Corollary 10.9a below, and Lemma 12.1... | Lemma | 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 | Mbeta_Hall_G | |
Mbeta_quo_nil : nilpotent (M^`(1) / M`_\beta). Proof. have /and3P[_ bMb b'M'Mb] := pHall_subl Mbeta_der1 sM'M Mbeta_Hall. apply: nilpotentS (Fitting_nil (M^`(1) / M`_\beta)) => /=. rewrite -{1}[_ / _]Sylow_gen gen_subG. apply/bigcupsP=> Q /SylowP[q _ /and3P[sQM' qQ _]]. apply: subset_trans (pcore_sub q _). rewrite p_co... | Lemma | 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 | Mbeta_quo_nil | |
beta'_der1_nil H : \beta(M)^'.-group H -> H \subset M^`(1) -> nilpotent H. Proof. move=> b'H sHM'; have [_ bMb _] := and3P Mbeta_Hall. have{b'H} tiMbH: M`_\beta :&: H = 1 := coprime_TIg (pnat_coprime bMb b'H). rewrite {tiMbH}(isog_nil (quotient_isog (subset_trans sHM' nMbM') tiMbH)). exact: nilpotentS (quotientS _ sHM'... | Lemma | 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'_der1_nil | |
beta'_cent_Sylow p q X : p \notin \beta(M) -> q \notin \beta(M) -> q.-group X -> (p != q) && (X \subset M^`(1)) || (p < q) && (X \subset M) -> [/\ (*a1*) exists2 P, p.-Sylow(M`_\sigma) (gval P) & X \subset 'C(P), (*a2*) p \in \alpha(M) -> 'C_M(X)%G \in 'U & (*a3*) q.-Sylow(M^`(1)) X -> exists2 P, p.-Sylow(M^`(1)) (gval... | Corollary | 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'_cent_Sylow | |
nonuniq_norm_Sylow_pprod p H S : H \in 'M -> H :!=: M -> p.-Sylow(G) S -> 'N(S) \subset H :&: M -> M`_\beta * (H :&: M) = M /\ \alpha(M) =i \beta(M). Proof. move=> maxH neqHM sylS_G sN_HM; have [sNH sNM] := subsetIP sN_HM. have [sSM sSH] := (subset_trans (normG S) sNM, subset_trans (normG S) sNH). have [sylS pS] := (pH... | Corollary | 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 | nonuniq_norm_Sylow_pprod | |
max_normed_2Elem_signaliser p q (A Q : {group gT}) : p != q -> A \in 'E_p^2(G) :&: 'E*_p(G) -> Q \in |/|*(A; q) -> q %| #|'C(A)| -> exists2 P : {group gT}, p.-Sylow(G) P /\ A \subset P & [/\ (*a*) 'O_p^'('C(P)) * ('N(P) :&: 'N(Q)) = 'N(P), (*b*) P \subset 'N(Q)^`(1) & (*c*) q.-narrow Q -> P \subset 'C(Q)]. Proof. move=... | Proposition | 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 | max_normed_2Elem_signaliser | |
sigma'_not_uniq K : K \subset M -> sigma'.-group K -> K \notin 'U. Proof. move=> sKM sg'K; have [E hallE sKE] := Hall_superset solM sKM sg'K. have [sEM sg'E _] := and3P hallE. have rEle2: 'r(E) <= 2. have [q _ ->] := rank_witness E; rewrite leqNgt; apply/negP=> rEgt2. have: q \in sigma' by rewrite (pnatPpi sg'E) // -p_... | Proposition | 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'_not_uniq | |
sub'cent_sigma_rank1 K : K \subset M -> sigma'.-group K -> 'r('C_K(M`_\sigma)) <= 1. Proof. move=> sKM sg'K; rewrite leqNgt; apply/rank_geP=> [[A /nElemP[p Ep2A]]]. have p_pr := pnElem_prime Ep2A. have [sACKMs abelA dimA] := pnElemP Ep2A; rewrite subsetI centsC in sACKMs. have{sACKMs} [sAK cAMs]: A \subset K /\ M`_\sig... | Proposition | 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 | sub'cent_sigma_rank1 | |
sub'cent_sigma_cyclic K (Y := 'C_K(M`_\sigma) :&: M^`(1)) : K \subset M -> sigma'.-group K -> cyclic Y /\ Y <| M. Proof. move=> sKM sg'K; pose Z := 'O_sigma'('F(M)). have nsZM: Z <| M by rewrite !gFnormal_trans. have [sZM nZM] := andP nsZM; have Fnil := Fitting_nil M. have rZle1: 'r(Z) <= 1. apply: leq_trans (rankS _) ... | Proposition | 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 | sub'cent_sigma_cyclic | |
commG_sigma'_1Elem_cyclic p K P (K0 := [~: K, P]) : K \subset M -> sigma'.-group K -> p \in sigma' -> P \in 'E_p^1('N_M(K)) -> 'C_(M`_\sigma)(P) = 1 -> p^'.-group K -> abelian K -> [/\ K0 \subset 'C(M`_\sigma), cyclic K0 & K0 <| M]. Proof. move=> sKM sg'K sg'p EpP regP p'K cKK. have nK0P: P \subset 'N(K0) := commg_norm... | Proposition | 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 | commG_sigma'_1Elem_cyclic | |
sigma_disjoint M H : M \in 'M -> H \in 'M -> gval H \notin M :^: G -> [/\ (*a*) M`_\alpha :&: H`_\sigma = 1, [predI \alpha(M) & \sigma(H)] =i pred0 & (*b*) nilpotent M`_\sigma -> M`_\sigma :&: H`_\sigma = 1 /\ [predI \sigma(M) & \sigma(H)] =i pred0]. Proof. move=> maxM maxH notjMH. suffices sigmaMHnil p: p \in [predI \... | Lemma | 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_disjoint | |
basic_p2maxElem_structure p A P : A \in 'E_p^2(G) :&: 'E*_p(G) -> p.-group P -> A \subset P -> ~~ abelian P -> let Z0 := ('Ohm_1('Z(P)))%G in [/\ (*a*) Z0 \in 'E_p^1(A), (*b*) exists Y : {group gT}, [/\ cyclic Y, Z0 \subset Y & forall A0, A0 \in 'E_p^1(A) :\ Z0 -> A0 \x Y = 'C_P(A)] & (*c*) [transitive 'N_P(A), on 'E_p... | Lemma | 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 | basic_p2maxElem_structure | |
beta_not_narrow p : p \in \beta(G) -> [disjoint 'E_p^2(G) & 'E*_p(G)] /\ (forall P, p.-Sylow(G) P -> [disjoint 'E_p^2(P) & 'E*_p(P)]). Proof. move/forall_inP=> nnG. have nnSyl P: p.-Sylow(G) P -> [disjoint 'E_p^2(P) & 'E*_p(P)]. by move/nnG; rewrite negb_imply negbK setI_eq0 => /andP[]. split=> //; apply/pred0Pn=> [[E ... | Proposition | 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_not_narrow | |
beta_noncyclic_uniq p R : p \in \beta(G) -> p.-group R -> 'r(R) > 1 -> R \in 'U. Proof. move=> b_p pR rRgt1; have [P sylP sRP] := Sylow_superset (subsetT R) pR. rewrite (rank_pgroup pR) in rRgt1; have [A Ep2A] := p_rank_geP rRgt1. have [sAR abelA dimA] := pnElemP Ep2A; have p_pr := pnElem_prime Ep2A. case: (pickP [pred... | Proposition | 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_noncyclic_uniq | |
beta_subnorm_uniq p P X : p \in \beta(G) -> p.-Sylow(G) P -> X \subset P -> 'N_P(X)%G \in 'U. Proof. move=> b_p sylP sXP; set Q := 'N_P(X)%G. have pP := pHall_pgroup sylP; have pQ: p.-group Q := pgroupS (subsetIl _ _) pP. have [| rQle1] := ltnP 1 'r(Q); first exact: beta_noncyclic_uniq pQ. have cycQ: cyclic Q. by rewri... | Proposition | 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_subnorm_uniq | |
beta_norm_sub_mmax M Y : M \in 'M -> \beta(M).-subgroup(M) Y -> Y :!=: 1 -> 'N(Y) \subset M. Proof. move=> maxM /andP[sYM bY] ntY. have [F1 | [q q_pr q_dv_FY]] := trivgVpdiv 'F(Y). by rewrite -(trivg_Fitting (solvableS sYM (mmax_sol maxM))) F1 eqxx in ntY. pose X := 'O_q(Y); have qX: q.-group X := pcore_pgroup q _. hav... | Proposition | 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_norm_sub_mmax | |
exceptional_FTmaximal := [/\ p \in \sigma(M)^', A \in 'E_p^2(M), A0 \in 'E_p^1(A) & 'N(A0) \subset M]. Hypotheses (maxM : M \in 'M) (excM : exceptional_FTmaximal). Hypotheses (sylP : p.-Sylow(M) P) (sAP : A \subset P). (* Splitting the excM hypothesis. *) | 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/BGsection11.v | exceptional_FTmaximal | |
sM'p : p \in \sigma(M)^'. Proof. by case: excM. Qed. | Let | 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/BGsection11.v | sM'p | |
Ep2A : A \in 'E_p^2(M). Proof. by case: excM. Qed. | Let | 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/BGsection11.v | Ep2A | |
Ep1A0 : A0 \in 'E_p^1(A). Proof. by case: excM. Qed. | Let | 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/BGsection11.v | Ep1A0 | |
sNA0_M : 'N(A0) \subset M. Proof. by case: excM. Qed. (* Arithmetics of p. *) | Let | 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/BGsection11.v | sNA0_M | |
p_pr : prime p := pnElem_prime Ep2A. | Let | 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/BGsection11.v | p_pr | |
p_gt1 : p > 1 := prime_gt1 p_pr. | Let | 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/BGsection11.v | p_gt1 | |
p_gt0 : p > 0 := prime_gt0 p_pr. (* Group orders. *) | Let | 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/BGsection11.v | p_gt0 | |
oA : #|A| = (p ^ 2)%N := card_pnElem Ep2A. | Let | 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/BGsection11.v | oA | |
oA0 : #|A0| = p := card_pnElem Ep1A0. (* Structure of A. *) | Let | 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/BGsection11.v | oA0 | |
abelA : p.-abelem A. Proof. by case/pnElemP: Ep2A. Qed. | Let | 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/BGsection11.v | abelA | |
pA : p.-group A := abelem_pgroup abelA. | Let | 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/BGsection11.v | pA | |
cAA : abelian A := abelem_abelian abelA. (* Various set inclusions. *) | Let | 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/BGsection11.v | cAA |
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