# math-comp/math-comp

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 (* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *) (* Distributed under the terms of CeCILL-B. *) Require Import mathcomp.ssreflect.ssreflect. From mathcomp Require Import ssrbool ssrfun eqtype ssrnat seq div fintype prime. From mathcomp Require Import bigop finset fingroup morphism automorphism quotient action. From mathcomp Require Import cyclic gproduct gfunctor commutator pgroup center nilpotent. (******************************************************************************) (* The Sylow theorem and its consequences, including the Frattini argument, *) (* the nilpotence of p-groups, and the Baer-Suzuki theorem. *) (* This file also defines: *) (* Zgroup G == G is a Z-group, i.e., has only cyclic Sylow p-subgroups. *) (******************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Import GroupScope. (* The mod p lemma for the action of p-groups. *) Section ModP. Variable (aT : finGroupType) (sT : finType) (D : {group aT}). Variable to : action D sT. Lemma pgroup_fix_mod (p : nat) (G : {group aT}) (S : {set sT}) : p.-group G -> [acts G, on S | to] -> #|S| = #|'Fix_(S | to)(G)| %[mod p]. Proof. move=> pG nSG; have sGD: G \subset D := acts_dom nSG. apply/eqP; rewrite -(cardsID 'Fix_to(G)) eqn_mod_dvd (leq_addr, addKn) //. have: [acts G, on S :\: 'Fix_to(G) | to]; last move/acts_sum_card_orbit <-. rewrite actsD // -(setIidPr sGD); apply: subset_trans (acts_subnorm_fix _ _). by rewrite setIS ?normG. apply: dvdn_sum => _ /imsetP[x /setDP[_ nfx] ->]. have [k oGx]: {k | #|orbit to G x| = (p ^ k)%N}. by apply: p_natP; apply: pnat_dvd pG; rewrite card_orbit_in ?dvdn_indexg. case: k oGx => [/card_orbit1 fix_x | k ->]; last by rewrite expnS dvdn_mulr. by case/afixP: nfx => a Ga; apply/set1P; rewrite -fix_x mem_orbit. Qed. End ModP. Section ModularGroupAction. Variables (aT rT : finGroupType) (D : {group aT}) (R : {group rT}). Variables (to : groupAction D R) (p : nat). Implicit Types (G H : {group aT}) (M : {group rT}). Lemma nontrivial_gacent_pgroup G M : p.-group G -> p.-group M -> {acts G, on group M | to} -> M :!=: 1 -> 'C_(M | to)(G) :!=: 1. Proof. move=> pG pM [nMG sMR] ntM; have [p_pr p_dv_M _] := pgroup_pdiv pM ntM. rewrite -cardG_gt1 (leq_trans (prime_gt1 p_pr)) 1?dvdn_leq ?cardG_gt0 //= /dvdn. by rewrite gacentE ?(acts_dom nMG) // setIA (setIidPl sMR) -pgroup_fix_mod. Qed. Lemma pcore_sub_astab_irr G M : p.-group M -> M \subset R -> acts_irreducibly G M to -> 'O_p(G) \subset 'C_G(M | to). Proof. move=> pM sMR /mingroupP[/andP[ntM nMG] minM]. have /andP[sGpG nGpG]: 'O_p(G) <| G := gFnormal _ G. have sGD := acts_dom nMG; have sGpD: 'O_p(G) \subset D := gFsub_trans _ sGD. rewrite subsetI sGpG -gacentC //=; apply/setIidPl; apply: minM (subsetIl _ _). rewrite nontrivial_gacent_pgroup ?pcore_pgroup //=; last first. by split; rewrite ?gFsub_trans. by apply: subset_trans (acts_subnorm_subgacent sGpD nMG); rewrite subsetI subxx. Qed. Lemma pcore_faithful_irr_act G M : p.-group M -> M \subset R -> acts_irreducibly G M to -> [faithful G, on M | to] -> 'O_p(G) = 1. Proof. move=> pM sMR irrG ffulG; apply/trivgP; apply: subset_trans ffulG. exact: pcore_sub_astab_irr. Qed. End ModularGroupAction. Section Sylow. Variables (p : nat) (gT : finGroupType) (G : {group gT}). Implicit Types P Q H K : {group gT}. Theorem Sylow's_theorem : [/\ forall P, [max P | p.-subgroup(G) P] = p.-Sylow(G) P, [transitive G, on 'Syl_p(G) | 'JG], forall P, p.-Sylow(G) P -> #|'Syl_p(G)| = #|G : 'N_G(P)| & prime p -> #|'Syl_p(G)| %% p = 1%N]. Proof. pose maxp A P := [max P | p.-subgroup(A) P]; pose S := [set P | maxp G P]. pose oG := orbit 'JG%act G. have actS: [acts G, on S | 'JG]. apply/subsetP=> x Gx; rewrite 3!inE; apply/subsetP=> P; rewrite 3!inE. exact: max_pgroupJ. have S_pG P: P \in S -> P \subset G /\ p.-group P. by rewrite inE => /maxgroupp/andP[]. have SmaxN P Q: Q \in S -> Q \subset 'N(P) -> maxp 'N_G(P) Q. rewrite inE => /maxgroupP[/andP[sQG pQ] maxQ] nPQ. apply/maxgroupP; rewrite /psubgroup subsetI sQG nPQ. by split=> // R; rewrite subsetI -andbA andbCA => /andP[_]; apply: maxQ. have nrmG P: P \subset G -> P <| 'N_G(P). by move=> sPG; rewrite /normal subsetIr subsetI sPG normG. have sylS P: P \in S -> p.-Sylow('N_G(P)) P. move=> S_P; have [sPG pP] := S_pG P S_P. by rewrite normal_max_pgroup_Hall ?nrmG //; apply: SmaxN; rewrite ?normG. have{SmaxN} defCS P: P \in S -> 'Fix_(S |'JG)(P) = [set P]. move=> S_P; apply/setP=> Q; rewrite {1}in_setI {1}afixJG. apply/andP/set1P=> [[S_Q nQP]|->{Q}]; last by rewrite normG. apply/esym/val_inj; case: (S_pG Q) => //= sQG _. by apply: uniq_normal_Hall (SmaxN Q _ _ _) => //=; rewrite ?sylS ?nrmG. have{defCS} oG_mod: {in S &, forall P Q, #|oG P| = (Q \in oG P) %[mod p]}. move=> P Q S_P S_Q; have [sQG pQ] := S_pG _ S_Q. have soP_S: oG P \subset S by rewrite acts_sub_orbit. have /pgroup_fix_mod-> //: [acts Q, on oG P | 'JG]. apply/actsP=> x /(subsetP sQG) Gx R; apply: orbit_transl. exact: mem_orbit. rewrite -{1}(setIidPl soP_S) -setIA defCS // (cardsD1 Q) setDE. by rewrite -setIA setICr setI0 cards0 addn0 inE set11 andbT. have [P S_P]: exists P, P \in S. have: p.-subgroup(G) 1 by rewrite /psubgroup sub1G pgroup1. by case/(@maxgroup_exists _ (p.-subgroup(G))) => P; exists P; rewrite inE. have trS: [transitive G, on S | 'JG]. apply/imsetP; exists P => //; apply/eqP. rewrite eqEsubset andbC acts_sub_orbit // S_P; apply/subsetP=> Q S_Q. have:= S_P; rewrite inE => /maxgroupP[/andP[_ pP]]. have [-> max1 | ntP _] := eqVneq P 1%G. move/andP/max1: (S_pG _ S_Q) => Q1. by rewrite (group_inj (Q1 (sub1G Q))) orbit_refl. have:= oG_mod _ _ S_P S_P; rewrite (oG_mod _ Q) // orbit_refl. have p_gt1: p > 1 by apply: prime_gt1; case/pgroup_pdiv: pP. by case: (Q \in oG P) => //; rewrite mod0n modn_small. have oS1: prime p -> #|S| %% p = 1%N. move/prime_gt1 => p_gt1. by rewrite -(atransP trS P S_P) (oG_mod P P) // orbit_refl modn_small. have oSiN Q: Q \in S -> #|S| = #|G : 'N_G(Q)|. by move=> S_Q; rewrite -(atransP trS Q S_Q) card_orbit astab1JG. have sylP: p.-Sylow(G) P. rewrite pHallE; case: (S_pG P) => // -> /= pP. case p_pr: (prime p); last first. rewrite p_part lognE p_pr /= -trivg_card1; apply/idPn=> ntP. by case/pgroup_pdiv: pP p_pr => // ->. rewrite -(LagrangeI G 'N(P)) /= mulnC partnM ?cardG_gt0 // part_p'nat. by rewrite mul1n (card_Hall (sylS P S_P)). by rewrite p'natE // -indexgI -oSiN // /dvdn oS1. have eqS Q: maxp G Q = p.-Sylow(G) Q. apply/idP/idP=> [S_Q|]; last exact: Hall_max. have{S_Q} S_Q: Q \in S by rewrite inE. rewrite pHallE -(card_Hall sylP); case: (S_pG Q) => // -> _ /=. by case: (atransP2 trS S_P S_Q) => x _ ->; rewrite cardJg. have ->: 'Syl_p(G) = S by apply/setP=> Q; rewrite 2!inE. by split=> // Q sylQ; rewrite -oSiN ?inE ?eqS. Qed. Lemma max_pgroup_Sylow P : [max P | p.-subgroup(G) P] = p.-Sylow(G) P. Proof. by case Sylow's_theorem. Qed. Lemma Sylow_superset Q : Q \subset G -> p.-group Q -> {P : {group gT} | p.-Sylow(G) P & Q \subset P}. Proof. move=> sQG pQ. have [|P] := @maxgroup_exists _ (p.-subgroup(G)) Q; first exact/andP. by rewrite max_pgroup_Sylow; exists P. Qed. Lemma Sylow_exists : {P : {group gT} | p.-Sylow(G) P}. Proof. by case: (Sylow_superset (sub1G G) (pgroup1 _ p)) => P; exists P. Qed. Lemma Syl_trans : [transitive G, on 'Syl_p(G) | 'JG]. Proof. by case Sylow's_theorem. Qed. Lemma Sylow_trans P Q : p.-Sylow(G) P -> p.-Sylow(G) Q -> exists2 x, x \in G & Q :=: P :^ x. Proof. move=> sylP sylQ; have:= (atransP2 Syl_trans) P Q; rewrite !inE. by case=> // x Gx ->; exists x. Qed. Lemma Sylow_subJ P Q : p.-Sylow(G) P -> Q \subset G -> p.-group Q -> exists2 x, x \in G & Q \subset P :^ x. Proof. move=> sylP sQG pQ; have [Px sylPx] := Sylow_superset sQG pQ. by have [x Gx ->] := Sylow_trans sylP sylPx; exists x. Qed. Lemma Sylow_Jsub P Q : p.-Sylow(G) P -> Q \subset G -> p.-group Q -> exists2 x, x \in G & Q :^ x \subset P. Proof. move=> sylP sQG pQ; have [x Gx] := Sylow_subJ sylP sQG pQ. by exists x^-1; rewrite (groupV, sub_conjgV). Qed. Lemma card_Syl P : p.-Sylow(G) P -> #|'Syl_p(G)| = #|G : 'N_G(P)|. Proof. by case: Sylow's_theorem P. Qed. Lemma card_Syl_dvd : #|'Syl_p(G)| %| #|G|. Proof. by case Sylow_exists => P /card_Syl->; apply: dvdn_indexg. Qed. Lemma card_Syl_mod : prime p -> #|'Syl_p(G)| %% p = 1%N. Proof. by case Sylow's_theorem. Qed. Lemma Frattini_arg H P : G <| H -> p.-Sylow(G) P -> G * 'N_H(P) = H. Proof. case/andP=> sGH nGH sylP; rewrite -normC ?subIset ?nGH ?orbT // -astab1JG. move/subgroup_transitiveP: Syl_trans => ->; rewrite ?inE //. apply/imsetP; exists P; rewrite ?inE //. apply/eqP; rewrite eqEsubset -{1}((atransP Syl_trans) P) ?inE // imsetS //=. by apply/subsetP=> _ /imsetP[x Hx ->]; rewrite inE -(normsP nGH x Hx) pHallJ2. Qed. End Sylow. Section MoreSylow. Variables (gT : finGroupType) (p : nat). Implicit Types G H P : {group gT}. Lemma Sylow_setI_normal G H P : G <| H -> p.-Sylow(H) P -> p.-Sylow(G) (G :&: P). Proof. case/normalP=> sGH nGH sylP; have [Q sylQ] := Sylow_exists p G. have /maxgroupP[/andP[sQG pQ] maxQ] := Hall_max sylQ. have [R sylR sQR] := Sylow_superset (subset_trans sQG sGH) pQ. have [[x Hx ->] pR] := (Sylow_trans sylR sylP, pHall_pgroup sylR). rewrite -(nGH x Hx) -conjIg pHallJ2. have /maxQ-> //: Q \subset G :&: R by rewrite subsetI sQG. by rewrite /psubgroup subsetIl (pgroupS _ pR) ?subsetIr. Qed. Lemma normal_sylowP G : reflect (exists2 P : {group gT}, p.-Sylow(G) P & P <| G) (#|'Syl_p(G)| == 1%N). Proof. apply: (iffP idP) => [syl1 | [P sylP nPG]]; last first. by rewrite (card_Syl sylP) (setIidPl _) (indexgg, normal_norm). have [P sylP] := Sylow_exists p G; exists P => //. rewrite /normal (pHall_sub sylP); apply/setIidPl; apply/eqP. rewrite eqEcard subsetIl -(LagrangeI G 'N(P)) -indexgI /=. by rewrite -(card_Syl sylP) (eqP syl1) muln1. Qed. Lemma trivg_center_pgroup P : p.-group P -> 'Z(P) = 1 -> P :=: 1. Proof. move=> pP Z1; apply/eqP/idPn=> ntP. have{ntP} [p_pr p_dv_P _] := pgroup_pdiv pP ntP. suff: p %| #|'Z(P)| by rewrite Z1 cards1 gtnNdvd ?prime_gt1. by rewrite /center /dvdn -afixJ -pgroup_fix_mod // astabsJ normG. Qed. Lemma p2group_abelian P : p.-group P -> logn p #|P| <= 2 -> abelian P. Proof. move=> pP lePp2; pose Z := 'Z(P); have sZP: Z \subset P := center_sub P. case: (eqVneq Z 1); first by move/(trivg_center_pgroup pP)->; apply: abelian1. case/(pgroup_pdiv (pgroupS sZP pP)) => p_pr _ [k oZ]. apply: cyclic_center_factor_abelian. have [->|] := eqVneq (P / Z) 1; first exact: cyclic1. have pPq := quotient_pgroup 'Z(P) pP; case/(pgroup_pdiv pPq) => _ _ [j oPq]. rewrite prime_cyclic // oPq; case: j oPq lePp2 => //= j. rewrite card_quotient ?gFnorm //. by rewrite -(Lagrange sZP) lognM // => ->; rewrite oZ !pfactorK ?addnS. Qed. Lemma card_p2group_abelian P : prime p -> #|P| = (p ^ 2)%N -> abelian P. Proof. move=> primep oP; have pP: p.-group P by rewrite /pgroup oP pnat_exp pnat_id. by rewrite (p2group_abelian pP) // oP pfactorK. Qed. Lemma Sylow_transversal_gen (T : {set {group gT}}) G : (forall P, P \in T -> P \subset G) -> (forall p, p \in \pi(G) -> exists2 P, P \in T & p.-Sylow(G) P) -> << \bigcup_(P in T) P >> = G. Proof. move=> G_T T_G; apply/eqP; rewrite eqEcard gen_subG. apply/andP; split; first exact/bigcupsP. apply: dvdn_leq (cardG_gt0 _) _; apply/dvdn_partP=> // q /T_G[P T_P sylP]. by rewrite -(card_Hall sylP); apply: cardSg; rewrite sub_gen // bigcup_sup. Qed. Lemma Sylow_gen G : <<\bigcup_(P : {group gT} | Sylow G P) P>> = G. Proof. set T := [set P : {group gT} | Sylow G P]. rewrite -{2}(@Sylow_transversal_gen T G) => [|P | q _]. - by congr <<_>>; apply: eq_bigl => P; rewrite inE. - by rewrite inE => /and3P[]. by case: (Sylow_exists q G) => P sylP; exists P; rewrite // inE (p_Sylow sylP). Qed. End MoreSylow. Section SomeHall. Variable gT : finGroupType. Implicit Types (p : nat) (pi : nat_pred) (G H K P R : {group gT}). Lemma Hall_pJsub p pi G H P : pi.-Hall(G) H -> p \in pi -> P \subset G -> p.-group P -> exists2 x, x \in G & P :^ x \subset H. Proof. move=> hallH pi_p sPG pP. have [S sylS] := Sylow_exists p H; have sylS_G := subHall_Sylow hallH pi_p sylS. have [x Gx sPxS] := Sylow_Jsub sylS_G sPG pP; exists x => //. exact: subset_trans sPxS (pHall_sub sylS). Qed. Lemma Hall_psubJ p pi G H P : pi.-Hall(G) H -> p \in pi -> P \subset G -> p.-group P -> exists2 x, x \in G & P \subset H :^ x. Proof. move=> hallH pi_p sPG pP; have [x Gx sPxH] := Hall_pJsub hallH pi_p sPG pP. by exists x^-1; rewrite ?groupV -?sub_conjg. Qed. Lemma Hall_setI_normal pi G K H : K <| G -> pi.-Hall(G) H -> pi.-Hall(K) (H :&: K). Proof. move=> nsKG hallH; have [sHG piH _] := and3P hallH. have [sHK_H sHK_K] := (subsetIl H K, subsetIr H K). rewrite pHallE sHK_K /= -(part_pnat_id (pgroupS sHK_H piH)); apply/eqP. rewrite (widen_partn _ (subset_leq_card sHK_K)); apply: eq_bigr => p pi_p. have [P sylP] := Sylow_exists p H. have sylPK := Sylow_setI_normal nsKG (subHall_Sylow hallH pi_p sylP). rewrite -!p_part -(card_Hall sylPK); symmetry; apply: card_Hall. by rewrite (pHall_subl _ sHK_K) //= setIC setSI ?(pHall_sub sylP). Qed. Lemma coprime_mulG_setI_norm H G K R : K * R = G -> G \subset 'N(H) -> coprime #|K| #|R| -> (K :&: H) * (R :&: H) = G :&: H. Proof. move=> defG nHG coKR; apply/eqP; rewrite eqEcard mulG_subG /= -defG. rewrite !setSI ?mulG_subl ?mulG_subr //=. rewrite coprime_cardMg ?(coKR, coprimeSg (subsetIl _ _), coprime_sym) //=. pose pi := \pi(K); have piK: pi.-group K by apply: pgroup_pi. have pi'R: pi^'.-group R by rewrite /pgroup -coprime_pi' /=. have [hallK hallR] := coprime_mulpG_Hall defG piK pi'R. have nsHG: H :&: G <| G by rewrite /normal subsetIr normsI ?normG. rewrite -!(setIC H) defG -(partnC pi (cardG_gt0 _)). rewrite -(card_Hall (Hall_setI_normal nsHG hallR)) /= setICA. rewrite -(card_Hall (Hall_setI_normal nsHG hallK)) /= setICA. by rewrite -defG (setIidPl (mulG_subl _ _)) (setIidPl (mulG_subr _ _)). Qed. End SomeHall. Section Nilpotent. Variable gT : finGroupType. Implicit Types (G H K P L : {group gT}) (p q : nat). Lemma pgroup_nil p P : p.-group P -> nilpotent P. Proof. move: {2}_.+1 (ltnSn #|P|) => n. elim: n gT P => // n IHn pT P; rewrite ltnS=> lePn pP. have [Z1 | ntZ] := eqVneq 'Z(P) 1. by rewrite (trivg_center_pgroup pP Z1) nilpotent1. rewrite -quotient_center_nil IHn ?morphim_pgroup // (leq_trans _ lePn) //. rewrite card_quotient ?normal_norm ?center_normal // -divgS ?subsetIl //. by rewrite ltn_Pdiv // ltnNge -trivg_card_le1. Qed. Lemma pgroup_sol p P : p.-group P -> solvable P. Proof. by move/pgroup_nil; apply: nilpotent_sol. Qed. Lemma small_nil_class G : nil_class G <= 5 -> nilpotent G. Proof. move=> leK5; case: (ltnP 5 #|G|) => [lt5G | leG5 {leK5}]. by rewrite nilpotent_class (leq_ltn_trans leK5). apply: pgroup_nil (pdiv #|G|) _ _; apply/andP; split=> //. by case: #|G| leG5 => //; do 5!case=> //. Qed. Lemma nil_class2 G : (nil_class G <= 2) = (G^`(1) \subset 'Z(G)). Proof. rewrite subsetI der_sub; apply/idP/commG1P=> [clG2 | L3G1]. by apply/(lcn_nil_classP 2); rewrite ?small_nil_class ?(leq_trans clG2). by apply/(lcn_nil_classP 2) => //; apply/lcnP; exists 2. Qed. Lemma nil_class3 G : (nil_class G <= 3) = ('L_3(G) \subset 'Z(G)). Proof. rewrite subsetI lcn_sub; apply/idP/commG1P=> [clG3 | L4G1]. by apply/(lcn_nil_classP 3); rewrite ?small_nil_class ?(leq_trans clG3). by apply/(lcn_nil_classP 3) => //; apply/lcnP; exists 3. Qed. Lemma nilpotent_maxp_normal pi G H : nilpotent G -> [max H | pi.-subgroup(G) H] -> H <| G. Proof. move=> nilG /maxgroupP[/andP[sHG piH] maxH]. have nHN: H <| 'N_G(H) by rewrite normal_subnorm. have{maxH} hallH: pi.-Hall('N_G(H)) H. apply: normal_max_pgroup_Hall => //; apply/maxgroupP. rewrite /psubgroup normal_sub // piH; split=> // K. by rewrite subsetI -andbA andbCA => /andP[_ /maxH]. rewrite /normal sHG; apply/setIidPl/esym. apply: nilpotent_sub_norm; rewrite ?subsetIl ?setIS //= char_norms //. by congr (_ \char _): (pcore_char pi 'N_G(H)); apply: normal_Hall_pcore. Qed. Lemma nilpotent_Hall_pcore pi G H : nilpotent G -> pi.-Hall(G) H -> H :=: 'O_pi(G). Proof. move=> nilG hallH; have maxH := Hall_max hallH; apply/eqP. rewrite eqEsubset pcore_max ?(pHall_pgroup hallH) //. by rewrite (normal_sub_max_pgroup maxH) ?pcore_pgroup ?pcore_normal. exact: nilpotent_maxp_normal maxH. Qed. Lemma nilpotent_pcore_Hall pi G : nilpotent G -> pi.-Hall(G) 'O_pi(G). Proof. move=> nilG; case: (@maxgroup_exists _ (psubgroup pi G) 1) => [|H maxH _]. by rewrite /psubgroup sub1G pgroup1. have hallH := normal_max_pgroup_Hall maxH (nilpotent_maxp_normal nilG maxH). by rewrite -(nilpotent_Hall_pcore nilG hallH). Qed. Lemma nilpotent_pcoreC pi G : nilpotent G -> 'O_pi(G) \x 'O_pi^'(G) = G. Proof. move=> nilG; have trO: 'O_pi(G) :&: 'O_pi^'(G) = 1. by apply: coprime_TIg; apply: (@pnat_coprime pi); apply: pcore_pgroup. rewrite dprodE //. apply/eqP; rewrite eqEcard mul_subG ?pcore_sub // (TI_cardMg trO). by rewrite !(card_Hall (nilpotent_pcore_Hall _ _)) // partnC ?leqnn. rewrite (sameP commG1P trivgP) -trO subsetI commg_subl commg_subr. by rewrite !gFsub_trans ?gFnorm. Qed. Lemma sub_nilpotent_cent2 H K G : nilpotent G -> K \subset G -> H \subset G -> coprime #|K| #|H| -> H \subset 'C(K). Proof. move=> nilG sKG sHG; rewrite coprime_pi' // => p'H. have sub_Gp := sub_Hall_pcore (nilpotent_pcore_Hall _ nilG). have [_ _ cGpp' _] := dprodP (nilpotent_pcoreC \pi(K) nilG). by apply: centSS cGpp'; rewrite sub_Gp ?pgroup_pi. Qed. Lemma pi_center_nilpotent G : nilpotent G -> \pi('Z(G)) = \pi(G). Proof. move=> nilG; apply/eq_piP => /= p. apply/idP/idP=> [|pG]; first exact: (piSg (center_sub _)). move: (pG); rewrite !mem_primes !cardG_gt0; case/andP=> p_pr _. pose Z := 'O_p(G) :&: 'Z(G); have ntZ: Z != 1. rewrite meet_center_nil ?pcore_normal // trivg_card_le1 -ltnNge. rewrite (card_Hall (nilpotent_pcore_Hall p nilG)) p_part. by rewrite (ltn_exp2l 0 _ (prime_gt1 p_pr)) logn_gt0. have pZ: p.-group Z := pgroupS (subsetIl _ _) (pcore_pgroup _ _). have{ntZ pZ} [_ pZ _] := pgroup_pdiv pZ ntZ. by rewrite p_pr (dvdn_trans pZ) // cardSg ?subsetIr. Qed. Lemma Sylow_subnorm p G P : p.-Sylow('N_G(P)) P = p.-Sylow(G) P. Proof. apply/idP/idP=> sylP; last first. apply: pHall_subl (subsetIl _ _) (sylP). by rewrite subsetI normG (pHall_sub sylP). have [/subsetIP[sPG sPN] pP _] := and3P sylP. have [Q sylQ sPQ] := Sylow_superset sPG pP; have [sQG pQ _] := and3P sylQ. rewrite -(nilpotent_sub_norm (pgroup_nil pQ) sPQ) {sylQ}//. rewrite subEproper eq_sym eqEcard subsetI sPQ sPN dvdn_leq //. rewrite -(part_pnat_id (pgroupS (subsetIl _ _) pQ)) (card_Hall sylP). by rewrite partn_dvd // cardSg ?setSI. Qed. End Nilpotent. Lemma nil_class_pgroup (gT : finGroupType) (p : nat) (P : {group gT}) : p.-group P -> nil_class P <= maxn 1 (logn p #|P|).-1. Proof. move=> pP; move def_c: (nil_class P) => c. elim: c => // c IHc in gT P def_c pP *; set e := logn p _. have nilP := pgroup_nil pP; have sZP := center_sub P. have [e_le2 | e_gt2] := leqP e 2. by rewrite -def_c leq_max nil_class1 (p2group_abelian pP). have pPq: p.-group (P / 'Z(P)) by apply: quotient_pgroup. rewrite -(subnKC e_gt2) ltnS (leq_trans (IHc _ _ _ pPq)) //. by rewrite nil_class_quotient_center ?def_c. rewrite geq_max /= -add1n -leq_subLR -subn1 -subnDA -subSS leq_sub2r //. rewrite ltn_log_quotient //= -(setIidPr sZP) meet_center_nil //. by rewrite -nil_class0 def_c. Qed. Definition Zgroup (gT : finGroupType) (A : {set gT}) := [forall (V : {group gT} | Sylow A V), cyclic V]. Section Zgroups. Variables (gT rT : finGroupType) (D : {group gT}) (f : {morphism D >-> rT}). Implicit Types G H K : {group gT}. Lemma ZgroupS G H : H \subset G -> Zgroup G -> Zgroup H. Proof. move=> sHG /forallP zgG; apply/forall_inP=> V /SylowP[p p_pr /and3P[sVH]]. case/(Sylow_superset (subset_trans sVH sHG))=> P sylP sVP _. by have:= zgG P; rewrite (p_Sylow sylP); apply: cyclicS. Qed. Lemma morphim_Zgroup G : Zgroup G -> Zgroup (f @* G). Proof. move=> zgG; wlog sGD: G zgG / G \subset D. by rewrite -morphimIdom; apply; rewrite (ZgroupS _ zgG, subsetIl) ?subsetIr. apply/forall_inP=> fV /SylowP[p pr_p sylfV]. have [P sylP] := Sylow_exists p G. have [|z _ ->] := @Sylow_trans p _ _ (f @* P)%G _ _ sylfV. by apply: morphim_pHall (sylP); apply: subset_trans (pHall_sub sylP) sGD. by rewrite cyclicJ morphim_cyclic ?(forall_inP zgG) //; apply/SylowP; exists p. Qed. Lemma nil_Zgroup_cyclic G : Zgroup G -> nilpotent G -> cyclic G. Proof. elim: {G}_.+1 {-2}G (ltnSn #|G|) => // n IHn G; rewrite ltnS => leGn ZgG nilG. have [->|[p pr_p pG]] := trivgVpdiv G; first by rewrite -cycle1 cycle_cyclic. have /dprodP[_ defG Cpp' _] := nilpotent_pcoreC p nilG. have /cyclicP[x def_p]: cyclic 'O_p(G). have:= forallP ZgG 'O_p(G)%G. by rewrite (p_Sylow (nilpotent_pcore_Hall p nilG)). have /cyclicP[x' def_p']: cyclic 'O_p^'(G). have sp'G := pcore_sub p^' G. apply: IHn (leq_trans _ leGn) (ZgroupS sp'G _) (nilpotentS sp'G _) => //. rewrite proper_card // properEneq sp'G andbT; case: eqP => //= def_p'. by have:= pcore_pgroup p^' G; rewrite def_p' /pgroup p'natE ?pG. apply/cyclicP; exists (x * x'); rewrite -{}defG def_p def_p' cycleM //. by red; rewrite -(centsP Cpp') // (def_p, def_p') cycle_id. by rewrite /order -def_p -def_p' (@pnat_coprime p) //; apply: pcore_pgroup. Qed. End Zgroups. Arguments Zgroup _ _%g. Prenex Implicits Zgroup. Section NilPGroups. Variables (p : nat) (gT : finGroupType). Implicit Type G P N : {group gT}. (* B & G 1.22 p.9 *) Lemma normal_pgroup r P N : p.-group P -> N <| P -> r <= logn p #|N| -> exists Q : {group gT}, [/\ Q \subset N, Q <| P & #|Q| = (p ^ r)%N]. Proof. elim: r gT P N => [|r IHr] gTr P N pP nNP le_r. by exists (1%G : {group gTr}); rewrite sub1G normal1 cards1. have [NZ_1 | ntNZ] := eqVneq (N :&: 'Z(P)) 1. by rewrite (TI_center_nil (pgroup_nil pP)) // cards1 logn1 in le_r. have: p.-group (N :&: 'Z(P)) by apply: pgroupS pP; rewrite /= setICA subsetIl. case/pgroup_pdiv=> // p_pr /Cauchy[// | z]. rewrite -cycle_subG !subsetI => /and3P[szN szP cPz] ozp _. have{cPz} nzP: P \subset 'N(<[z]>) by rewrite cents_norm // centsC. have: N / <[z]> <| P / <[z]> by rewrite morphim_normal. case/IHr=> [||Qb [sQNb nQPb]]; first exact: morphim_pgroup. rewrite card_quotient ?(subset_trans (normal_sub nNP)) // -ltnS. apply: (leq_trans le_r); rewrite -(Lagrange szN) [#|_|]ozp. by rewrite lognM // ?prime_gt0 // logn_prime ?eqxx. case/(inv_quotientN _): nQPb sQNb => [|Q -> szQ nQP]; first exact/andP. have nzQ := subset_trans (normal_sub nQP) nzP. rewrite quotientSGK // card_quotient // => sQN izQ. by exists Q; split=> //; rewrite expnS -izQ -ozp Lagrange. Qed. Theorem Baer_Suzuki x G : x \in G -> (forall y, y \in G -> p.-group <<[set x; x ^ y]>>) -> x \in 'O_p(G). Proof. elim: {G}_.+1 {-2}G x (ltnSn #|G|) => // n IHn G x; rewrite ltnS. set E := x ^: G => leGn Gx pE. have{pE} pE: {in E &, forall x1 x2, p.-group <<[set x1; x2]>>}. move=> _ _ /imsetP[y1 Gy1 ->] /imsetP[y2 Gy2 ->]. rewrite -(mulgKV y1 y2) conjgM -2!conjg_set1 -conjUg genJ pgroupJ. by rewrite pE // groupMl ?groupV. have sEG: <> \subset G by rewrite gen_subG class_subG. have nEG: G \subset 'N(E) by apply: class_norm. have Ex: x \in E by apply: class_refl. have [P Px sylP]: exists2 P : {group gT}, x \in P & p.-Sylow(<>) P. have sxxE: <<[set x; x]>> \subset <> by rewrite genS // setUid sub1set. have{sxxE} [P sylP sxxP] := Sylow_superset sxxE (pE _ _ Ex Ex). by exists P => //; rewrite (subsetP sxxP) ?mem_gen ?setU11. case sEP: (E \subset P). apply: subsetP Ex; rewrite -gen_subG; apply: pcore_max. by apply: pgroupS (pHall_pgroup sylP); rewrite gen_subG. by rewrite /normal gen_subG class_subG // norms_gen. pose P_yD D := [pred y in E :\: P | p.-group <>]. pose P_D := [pred D : {set gT} | D \subset P :&: E & [exists y, P_yD D y]]. have{Ex Px}: P_D [set x]. rewrite /= sub1set inE Px Ex; apply/existsP=> /=. by case/subsetPn: sEP => y Ey Py; exists y; rewrite inE Ey Py pE. case/(@maxset_exists _ P_D)=> D /maxsetP[]; rewrite {P_yD P_D}/=. rewrite subsetI sub1set -andbA => /and3P[sDP sDE /existsP[y0]]. set B := _ |: D; rewrite inE -andbA => /and3P[Py0 Ey0 pB] maxD Dx. have sDgE: D \subset <> by apply: sub_gen. have sDG: D \subset G by apply: subset_trans sEG. have sBE: B \subset E by rewrite subUset sub1set Ey0. have sBG: <> \subset G by apply: subset_trans (genS _) sEG. have sDB: D \subset B by rewrite subsetUr. have defD: D :=: P :&: <> :&: E. apply/eqP; rewrite eqEsubset ?subsetI sDP sDE sub_gen //=. apply/setUidPl; apply: maxD; last apply: subsetUl. rewrite subUset subsetI sDP sDE setIAC subsetIl. apply/existsP; exists y0; rewrite inE Py0 Ey0 /= setUA -/B. by rewrite -[<<_>>]joing_idl joingE setKI genGid. have nDD: D \subset 'N(D). apply/subsetP=> z Dz; rewrite inE defD. apply/subsetP=> _ /imsetP[y /setIP[PBy Ey] ->]. rewrite inE groupJ // ?inE ?(subsetP sDP) ?mem_gen ?setU1r //= memJ_norm //. exact: (subsetP (subset_trans sDG nEG)). case nDG: (G \subset 'N(D)). apply: subsetP Dx; rewrite -gen_subG pcore_max ?(pgroupS (genS _) pB) //. by rewrite /normal gen_subG sDG norms_gen. have{n leGn IHn nDG} pN: p.-group <<'N_E(D)>>. apply: pgroupS (pcore_pgroup p 'N_G(D)); rewrite gen_subG /=. apply/subsetP=> x1 /setIP[Ex1 Nx1]; apply: IHn => [||y Ny]. - apply: leq_trans leGn; rewrite proper_card // /proper subsetIl. by rewrite subsetI nDG andbF. - by rewrite inE Nx1 (subsetP sEG) ?mem_gen. have Ex1y: x1 ^ y \in E. by rewrite -mem_conjgV (normsP nEG) // groupV; case/setIP: Ny. apply: pgroupS (genS _) (pE _ _ Ex1 Ex1y). by apply/subsetP=> u; rewrite !inE. have [y1 Ny1 Py1]: exists2 y1, y1 \in 'N_E(D) & y1 \notin P. case sNN: ('N_<>('N_<>(D)) \subset 'N_<>(D)). exists y0 => //; have By0: y0 \in <> by rewrite mem_gen ?setU11. rewrite inE Ey0 -By0 -in_setI. by rewrite -['N__(D)](nilpotent_sub_norm (pgroup_nil pB)) ?subsetIl. case/subsetPn: sNN => z /setIP[Bz NNz]; rewrite inE Bz inE. case/subsetPn=> y; rewrite mem_conjg => Dzy Dy. have:= Dzy; rewrite {1}defD; do 2![case/setIP]=> _ Bzy Ezy. have Ey: y \in E by rewrite -(normsP nEG _ (subsetP sBG z Bz)) mem_conjg. have /setIP[By Ny]: y \in 'N_<>(D). by rewrite -(normP NNz) mem_conjg inE Bzy ?(subsetP nDD). exists y; first by rewrite inE Ey. by rewrite defD 2!inE Ey By !andbT in Dy. have [y2 Ny2 Dy2]: exists2 y2, y2 \in 'N_(P :&: E)(D) & y2 \notin D. case sNN: ('N_P('N_P(D)) \subset 'N_P(D)). have [z /= Ez sEzP] := Sylow_Jsub sylP (genS sBE) pB. have Gz: z \in G by apply: subsetP Ez. have /subsetPn[y Bzy Dy]: ~~ (B :^ z \subset D). apply/negP; move/subset_leq_card; rewrite cardJg cardsU1. by rewrite {1}defD 2!inE (negPf Py0) ltnn. exists y => //; apply: subsetP Bzy. rewrite -setIA setICA subsetI sub_conjg (normsP nEG) ?groupV // sBE. have nilP := pgroup_nil (pHall_pgroup sylP). by rewrite -['N__(_)](nilpotent_sub_norm nilP) ?subsetIl // -gen_subG genJ. case/subsetPn: sNN => z /setIP[Pz NNz]; rewrite 2!inE Pz. case/subsetPn=> y Dzy Dy; exists y => //; apply: subsetP Dzy. rewrite -setIA setICA subsetI sub_conjg (normsP nEG) ?groupV //. by rewrite sDE -(normP NNz); rewrite conjSg subsetI sDP. by apply: subsetP Pz; apply: (subset_trans (pHall_sub sylP)). suff{Dy2} Dy2D: y2 |: D = D by rewrite -Dy2D setU11 in Dy2. apply: maxD; last by rewrite subsetUr. case/setIP: Ny2 => PEy2 Ny2; case/setIP: Ny1 => Ey1 Ny1. rewrite subUset sub1set PEy2 subsetI sDP sDE. apply/existsP; exists y1; rewrite inE Ey1 Py1; apply: pgroupS pN. rewrite genS // !subUset !sub1set !in_setI Ey1 Ny1. by case/setIP: PEy2 => _ ->; rewrite Ny2 subsetI sDE. Qed. End NilPGroups.