fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
norm_conj_centA G x : x \in 'C(A) ->
(A \subset 'N(G :^ x)) = (A \subset 'N(G)).
Proof. by move=> cAx; rewrite norm_conj_norm ?(subsetP (cent_sub A)). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
norm_conj_cent
| |
strongest_coprime_quotient_centA G H :
let R := H :&: [~: G, A] in
A \subset 'N(H) -> R \subset G -> coprime #|R| #|A| ->
solvable R || solvable A ->
'C_G(A) / H = 'C_(G / H)(A / H).
Proof.
move=> R nHA sRG coRA solRA.
have nRA: A \subset 'N(R) by rewrite normsI ?commg_normr.
apply/eqP; rewrite eqEsubset subsetI morphimS ?subsetIl //=.
rewrite (subset_trans _ (morphim_cent _ _)) ?morphimS ?subsetIr //=.
apply/subsetP=> _ /setIP[/morphimP[x Nx Gx ->] cAHx].
have{cAHx} cAxR y: y \in A -> [~ x, y] \in R.
move=> Ay; have Ny: y \in 'N(H) by apply: subsetP Ay.
rewrite inE mem_commg // andbT coset_idr ?groupR // morphR //=.
by apply/eqP; apply/commgP; apply: (centP cAHx); rewrite mem_quotient.
have AxRA: A :^ x \subset R * A.
apply/subsetP=> _ /imsetP[y Ay ->].
rewrite -normC // -(mulKVg y (y ^ x)) -commgEl mem_mulg //.
by rewrite -groupV invg_comm cAxR.
have [y Ry def_Ax]: exists2 y, y \in R & A :^ x = A :^ y.
have oAx: #|A :^ x| = #|A| by rewrite cardJg.
have [solR | solA] := orP solRA; first exact: SchurZassenhaus_trans_sol.
by apply: SchurZassenhaus_trans_actsol; rewrite // joingC norm_joinEr.
rewrite -imset_coset; apply/imsetP; exists (x * y^-1); last first.
by rewrite conjgCV mkerl // ker_coset memJ_norm groupV; case/setIP: Ry.
rewrite /= inE groupMl // ?(groupV, subsetP sRG) //=.
apply/centP=> z Az; apply/commgP/eqP/set1P.
rewrite -[[set 1]](coprime_TIg coRA) inE {1}commgEl commgEr /= -/R.
rewrite invMg -mulgA invgK (@groupMl _ R) // conjMg mulgA -commgEl.
rewrite gr
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
strongest_coprime_quotient_cent
| |
coprime_norm_quotient_centA G H :
A \subset 'N(G) -> A \subset 'N(H) -> coprime #|H| #|A| -> solvable H ->
'C_G(A) / H = 'C_(G / H)(A / H).
Proof.
move=> nGA nHA coHA solH; have sRH := subsetIl H [~: G, A].
rewrite strongest_coprime_quotient_cent ?(coprimeSg sRH) 1?(solvableS sRH) //.
by rewrite subIset // commg_subl nGA orbT.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
coprime_norm_quotient_cent
| |
coprime_cent_mulGA G H :
A \subset 'N(G) -> A \subset 'N(H) -> G \subset 'N(H) ->
coprime #|H| #|A| -> solvable H ->
'C_(H * G)(A) = 'C_H(A) * 'C_G(A).
Proof.
move=> nHA nGA nHG coHA solH; rewrite -norm_joinEr //.
have nsHG: H <| H <*> G by rewrite /normal joing_subl join_subG normG.
rewrite -{2}(setIidPr (normal_sub nsHG)) setIAC.
rewrite group_modr ?setSI ?joing_subr //=; symmetry; apply/setIidPl.
rewrite -quotientSK ?subIset 1?normal_norm //.
by rewrite !coprime_norm_quotient_cent ?normsY //= norm_joinEr ?quotientMidl.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
coprime_cent_mulG
| |
quotient_TI_subcentK G H :
G \subset 'N(K) -> G \subset 'N(H) -> K :&: H = 1 ->
'C_K(G) / H = 'C_(K / H)(G / H).
Proof.
move=> nGK nGH tiKH.
have tiHR: H :&: [~: K, G] = 1.
by apply/trivgP; rewrite /= setIC -tiKH setSI ?commg_subl.
apply: strongest_coprime_quotient_cent; rewrite ?tiHR ?sub1G ?solvable1 //.
by rewrite cards1 coprime1n.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
quotient_TI_subcent
| |
external_action_im_coprime: coprime #|G'| #|A'|.
Proof. by rewrite !card_injm. Qed.
Let coGA' := external_action_im_coprime.
Let solG' : solvable G' := morphim_sol _ solG.
Let nGA' := im_sdpair_norm to.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
external_action_im_coprime
| |
ext_coprime_Hall_exists:
exists2 H : {group gT}, pi.-Hall(G) H & [acts A, on H | to].
Proof.
have [H' hallH' nHA'] := coprime_Hall_exists pi nGA' coGA' solG'.
have sHG' := pHall_sub hallH'.
exists (inG @*^-1 H')%G => /=.
by rewrite -(morphim_invmE injG) -{1}(im_invm injG) morphim_pHall.
by rewrite actsEsd ?morphpreK // subsetIl.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
ext_coprime_Hall_exists
| |
ext_coprime_Hall_trans(H1 H2 : {group gT}) :
pi.-Hall(G) H1 -> [acts A, on H1 | to] ->
pi.-Hall(G) H2 -> [acts A, on H2 | to] ->
exists2 x, x \in 'C_(G | to)(A) & H1 :=: H2 :^ x.
Proof.
move=> hallH1 nH1A hallH2 nH2A.
have sH1G := pHall_sub hallH1; have sH2G := pHall_sub hallH2.
rewrite !actsEsd // in nH1A nH2A.
have hallH1': pi.-Hall(G') (inG @* H1) by rewrite morphim_pHall.
have hallH2': pi.-Hall(G') (inG @* H2) by rewrite morphim_pHall.
have [x'] := coprime_Hall_trans nGA' coGA' solG' hallH1' nH1A hallH2' nH2A.
case/setIP=> /= Gx' cAx' /eqP defH1; pose x := invm injG x'.
have Gx: x \in G by rewrite -(im_invm injG) mem_morphim.
have def_x': x' = inG x by rewrite invmK.
exists x; first by rewrite inE Gx gacentEsd mem_morphpre /= -?def_x'.
apply/eqP; move: defH1; rewrite def_x' /= -morphimJ //=.
by rewrite !eqEsubset !injmSK // conj_subG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
ext_coprime_Hall_trans
| |
ext_norm_conj_cent(H : {group gT}) x :
H \subset G -> x \in 'C_(G | to)(A) ->
[acts A, on H :^ x | to] = [acts A, on H | to].
Proof.
move=> sHG /setIP[Gx].
rewrite gacentEsd !actsEsd ?conj_subG ?morphimJ // 2!inE Gx /=.
exact: norm_conj_cent.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
ext_norm_conj_cent
| |
ext_coprime_Hall_subset(X : {group gT}) :
X \subset G -> pi.-group X -> [acts A, on X | to] ->
exists H : {group gT},
[/\ pi.-Hall(G) H, [acts A, on H | to] & X \subset H].
Proof.
move=> sXG piX; rewrite actsEsd // => nXA'.
case: (coprime_Hall_subset nGA' coGA' solG' _ (morphim_pgroup _ piX) nXA').
exact: morphimS.
move=> H' /= [piH' nHA' sXH']; have sHG' := pHall_sub piH'.
exists (inG @*^-1 H')%G; rewrite actsEsd ?subsetIl ?morphpreK // nHA'.
rewrite -sub_morphim_pre //= sXH'; split=> //.
by rewrite -(morphim_invmE injG) -{1}(im_invm injG) morphim_pHall.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
ext_coprime_Hall_subset
| |
ext_coprime_quotient_cent(H : {group gT}) :
H \subset G -> [acts A, on H | to] -> coprime #|H| #|A| -> solvable H ->
'C_(|to)(A) / H = 'C_(|to / H)(A).
Proof.
move=> sHG nHA coHA solH; pose N := 'N_G(H).
have nsHN: H <| N by rewrite normal_subnorm.
have [sHN nHn] := andP nsHN.
have sNG: N \subset G by apply: subsetIl.
have nNA: {acts A, on group N | to}.
split; rewrite // actsEsd // injm_subnorm ?injm_sdpair1 //=.
by rewrite normsI ?norms_norm ?im_sdpair_norm -?actsEsd.
rewrite -!(gacentIdom _ A) -quotientInorm -gacentIim setIAC.
rewrite -(gacent_actby nNA) gacentEsd -morphpreIim /= -/N.
have:= (injm_sdpair1 <[nNA]>, injm_sdpair2 <[nNA]>).
set inG := sdpair1 _; set inA := sdpair2 _ => [[injG injA]].
set G' := inG @* N; set A' := inA @* A; pose H' := inG @* H.
have defN: 'N(H | to) = A by apply/eqP; rewrite eqEsubset subsetIl.
have def_Dq: qact_dom to H = A by rewrite qact_domE.
have sAq: A \subset qact_dom to H by rewrite def_Dq.
rewrite {2}def_Dq -(gacent_ract _ sAq); set to_q := (_ \ _)%gact.
have:= And3 (sdprod_sdpair to_q) (injm_sdpair1 to_q) (injm_sdpair2 to_q).
rewrite gacentEsd; set inAq := sdpair2 _; set inGq := sdpair1 _ => /=.
set Gq := inGq @* _; set Aq := inAq @* _ => [[q_d iGq iAq]].
have nH': 'N(H') = setT.
apply/eqP; rewrite -subTset -im_sdpair mulG_subG morphim_norms //=.
by rewrite -actsEsd // acts_actby subxx /= (setIidPr sHN).
have: 'dom (coset H' \o inA \o invm iAq) = Aq.
by rewrite ['dom _]morphpre_invm /= nH' morphpreT.
case/domP=> /= qA [def_qA ker_qA _ im_qA].
h
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
ext_coprime_quotient_cent
| |
sol_coprime_Sylow_existsA G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
exists2 P : {group gT}, p.-Sylow(G) P & A \subset 'N(P).
Proof.
move=> solA nGA coGA; pose AG := A <*> G.
have nsG_AG: G <| AG by rewrite /normal joing_subr join_subG nGA normG.
have [sG_AG nG_AG]:= andP nsG_AG.
have [P sylP] := Sylow_exists p G; pose N := 'N_AG(P); pose NG := G :&: N.
have nGN: N \subset 'N(G) by rewrite subIset ?nG_AG.
have sNG_G: NG \subset G := subsetIl G N.
have nsNG_N: NG <| N by rewrite /normal subsetIr normsI ?normG.
have defAG: G * N = AG := Frattini_arg nsG_AG sylP.
have oA : #|A| = #|N| %/ #|NG|.
rewrite /NG setIC divgI -card_quotient // -quotientMidl defAG.
rewrite card_quotient -?divgS //= norm_joinEl //.
by rewrite coprime_cardMg 1?coprime_sym // mulnK.
have: [splits N, over NG].
rewrite SchurZassenhaus_split // /Hall -divgS subsetIr //.
by rewrite -oA (coprimeSg sNG_G).
case/splitsP=> B; case/complP=> tNG_B defN.
have [nPB]: B \subset 'N(P) /\ B \subset AG.
by apply/andP; rewrite andbC -subsetI -/N -defN mulG_subr.
case/SchurZassenhaus_trans_actsol => // [|x Gx defB].
by rewrite oA -defN TI_cardMg // mulKn.
exists (P :^ x^-1)%G; first by rewrite pHallJ ?groupV.
by rewrite normJ -sub_conjg -defB.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
sol_coprime_Sylow_exists
| |
sol_coprime_Sylow_transA G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
[transitive 'C_G(A), on [set P in 'Syl_p(G) | A \subset 'N(P)] | 'JG].
Proof.
move=> solA nGA coGA; pose AG := A <*> G; set FpA := finset _.
have nG_AG: AG \subset 'N(G) by rewrite join_subG nGA normG.
have [P sylP nPA] := sol_coprime_Sylow_exists solA nGA coGA.
pose N := 'N_AG(P); have sAN: A \subset N by rewrite subsetI joing_subl.
have trNPA: A :^: AG ::&: N = A :^: N.
pose NG := 'N_G(P); have sNG_G : NG \subset G := subsetIl _ _.
have nNGA: A \subset 'N(NG) by rewrite normsI ?norms_norm.
apply/setP=> Ax; apply/setIdP/imsetP=> [[]|[x Nx ->{Ax}]]; last first.
by rewrite conj_subG //; case/setIP: Nx => AGx; rewrite imset_f.
have ->: N = A <*> NG by rewrite /N /AG !norm_joinEl // -group_modl.
have coNG_A := coprimeSg sNG_G coGA; case/imsetP=> x AGx ->{Ax}.
case/SchurZassenhaus_trans_actsol; rewrite ?cardJg // => y Ny /= ->.
by exists y; rewrite // mem_gen 1?inE ?Ny ?orbT.
have{trNPA}: [transitive 'N_AG(A), on FpA | 'JG].
have ->: FpA = 'Fix_('Syl_p(G) | 'JG)(A).
by apply/setP=> Q; rewrite 4!inE afixJG.
have SylP : P \in 'Syl_p(G) by rewrite inE.
apply/(trans_subnorm_fixP _ SylP); rewrite ?astab1JG //.
rewrite (atrans_supgroup _ (Syl_trans _ _)) ?joing_subr //= -/AG.
by apply/actsP=> x /= AGx Q /=; rewrite !inE -{1}(normsP nG_AG x) ?pHallJ2.
rewrite {1}/AG norm_joinEl // -group_modl ?normG ?coprime_norm_cent //=.
rewrite -cent_joinEr ?subsetIr // => trC_FpA.
have FpA_P: P \in FpA
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
sol_coprime_Sylow_trans
| |
sol_coprime_Sylow_subsetA G X :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable A ->
X \subset G -> p.-group X -> A \subset 'N(X) ->
exists P : {group gT}, [/\ p.-Sylow(G) P, A \subset 'N(P) & X \subset P].
Proof.
move=> nGA coGA solA sXG pX nXA.
pose nAp (Q : {group gT}) := [&& p.-group Q, Q \subset G & A \subset 'N(Q)].
have: nAp X by apply/and3P.
case/maxgroup_exists=> R; case/maxgroupP; case/and3P=> pR sRG nRA maxR sXR.
have [P sylP sRP]:= Sylow_superset sRG pR.
suffices defP: P :=: R by exists P; rewrite sylP defP.
case/and3P: sylP => sPG pP _; apply: (nilpotent_sub_norm (pgroup_nil pP)) => //.
pose N := 'N_G(R); have{sPG} sPN_N: 'N_P(R) \subset N by apply: setSI.
apply: norm_sub_max_pgroup (pgroupS (subsetIl _ _) pP) sPN_N (subsetIr _ _).
have nNA: A \subset 'N(N) by rewrite normsI ?norms_norm.
have coNA: coprime #|N| #|A| by apply: coprimeSg coGA; rewrite subsetIl.
have{solA coNA} [Q sylQ nQA] := sol_coprime_Sylow_exists solA nNA coNA.
suffices defQ: Q :=: R by rewrite max_pgroup_Sylow -{2}defQ.
apply: maxR; first by apply/and3P; case/and3P: sylQ; rewrite subsetI; case/andP.
by apply: normal_sub_max_pgroup (Hall_max sylQ) pR _; rewrite normal_subnorm.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
sol_coprime_Sylow_subset
| |
section(gT : finGroupType) := GSection of {group gT} * {group gT}.
Delimit Scope section_scope with sec.
Bind Scope section_scope with section.
|
Inductive
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
section
| |
mkSec(gT : finGroupType) (G1 G2 : {group gT}) := GSection (G1, G2).
Infix "/" := mkSec : section_scope.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
mkSec
| |
pair_of_sectiongT (s : section gT) := let: GSection u := s in u.
|
Coercion
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
pair_of_section
| |
quotient_of_sectiongT (s : section gT) : GroupSet.sort _ := s.1 / s.2.
|
Coercion
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
quotient_of_section
| |
section_groupgT (s : section gT) : {group (coset_of s.2)} :=
Eval hnf in [group of s].
|
Coercion
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
section_group
| |
Definition_ := [isNew for (@pair_of_section gT)].
HB.instance Definition _ := [Finite of section gT by <:].
|
HB.instance
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
Definition
| |
section_group.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
section_group
| |
section_isog:= [rel x y : section gT | x \isog y].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
section_isog
| |
section_reprs := odflt (1 / 1)%sec (pick (section_isog ^~ s)).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
section_repr
| |
mksreprG1 G2 := section_repr (mkSec G1 G2).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
mksrepr
| |
section_reprPs : section_repr s \isog s.
Proof.
by rewrite /section_repr; case: pickP => //= /(_ s); rewrite isog_refl.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
section_reprP
| |
section_repr_isogs1 s2 :
s1 \isog s2 -> section_repr s1 = section_repr s2.
Proof.
by move=> iso12; congr (odflt _ _); apply: eq_pick => s; apply: isog_transr.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
section_repr_isog
| |
mkfactors(G : {group gT}) (s : seq {group gT}) :=
map section_repr (pairmap (@mkSec _) G s).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
mkfactors
| |
compsG s := ((last G s) == 1%G) && compo.-series G s.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
comps
| |
compsPG s :
reflect (last G s = 1%G /\ path [rel x y : gTg | maxnormal y x x] G s)
(comps G s).
Proof. by apply: (iffP andP) => [] [/eqP]. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
compsP
| |
trivg_compsG s : comps G s -> (G :==: 1) = (s == [::]).
Proof.
case/andP=> ls cs; apply/eqP/eqP=> [G1 | s1]; last first.
by rewrite s1 /= in ls; apply/eqP.
by case: s {ls} cs => //= H s /andP[/maxgroupp]; rewrite G1 /proper sub1G andbF.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
trivg_comps
| |
comps_consG H s : comps G (H :: s) -> comps H s.
Proof. by case/andP => /= ls /andP[_]; rewrite /comps ls. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
comps_cons
| |
simple_compsPG s : comps G s -> reflect (s = [:: 1%G]) (simple G).
Proof.
move=> cs; apply: (iffP idP) => [|s1]; last first.
by rewrite s1 /comps eqxx /= andbT -simple_maxnormal in cs.
case: s cs => [/trivg_comps/eqP-> | H s]; first by case/simpleP; rewrite eqxx.
rewrite [comps _ _]andbCA /= => /andP[/maxgroupp maxH /trivg_comps/esym nil_s].
rewrite simple_maxnormal => /maxgroupP[_ simG].
have H1: H = 1%G by apply/val_inj/simG; rewrite // sub1G.
by move: nil_s; rewrite H1 eqxx => /eqP->.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
simple_compsP
| |
exists_comps(G : gTg) : exists s, comps G s.
Proof.
elim: {G} #|G| {1 3}G (leqnn #|G|) => [G | n IHn G cG].
by rewrite leqNgt cardG_gt0.
have [sG | nsG] := boolP (simple G).
by exists [:: 1%G]; rewrite /comps eqxx /= -simple_maxnormal andbT.
have [-> | ntG] := eqVneq G 1%G; first by exists [::]; rewrite /comps eqxx.
have [N maxN] := ex_maxnormal_ntrivg ntG.
have [|s /andP[ls cs]] := IHn N.
by rewrite -ltnS (leq_trans _ cG) // proper_card // (maxnormal_proper maxN).
by exists (N :: s); apply/and3P.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
exists_comps
| |
JordanHolderUniqueness(G : gTg) (s1 s2 : seq gTg) :
comps G s1 -> comps G s2 -> perm_eq (mkfactors G s1) (mkfactors G s2).
Proof.
have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => /ltnSE-cG cs1 cs2.
have [G1 | ntG] := boolP (G :==: 1).
have -> : s1 = [::] by apply/eqP; rewrite -(trivg_comps cs1).
have -> : s2 = [::] by apply/eqP; rewrite -(trivg_comps cs2).
by rewrite /= perm_refl.
have [sG | nsG] := boolP (simple G).
by rewrite (simple_compsP cs1 sG) (simple_compsP cs2 sG) perm_refl.
case es1: s1 cs1 => [|N1 st1] cs1.
by move: (trivg_comps cs1); rewrite eqxx; move/negP:ntG.
case es2: s2 cs2 => [|N2 st2] cs2 {s1 es1}.
by move: (trivg_comps cs2); rewrite eqxx; move/negP:ntG.
case/andP: cs1 => /= lst1; case/andP=> maxN_1 pst1.
case/andP: cs2 => /= lst2; case/andP=> maxN_2 pst2.
have cN1 : #|N1| < n.
by rewrite (leq_trans _ cG) ?proper_card ?(maxnormal_proper maxN_1).
have cN2 : #|N2| < n.
by rewrite (leq_trans _ cG) ?proper_card ?(maxnormal_proper maxN_2).
case: (N1 =P N2) {s2 es2} => [eN12 |].
by rewrite eN12 /= perm_cons Hi // /comps ?lst2 //= -eN12 lst1.
move/eqP; rewrite -val_eqE /=; move/eqP=> neN12.
have nN1G : N1 <| G by apply: maxnormal_normal.
have nN2G : N2 <| G by apply: maxnormal_normal.
pose N := (N1 :&: N2)%G.
have nNG : N <| G.
by rewrite /normal subIset ?(normal_sub nN1G) //= normsI ?normal_norm.
have iso1 : (G / N1)%G \isog (N2 / N)%G.
rewrite isog_sym /= -(maxnormalM maxN_1 maxN_2) //.
rewrite (@normC _ N1 N2) ?(subset_trans (normal_sub nN1G)) ?n
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
JordanHolderUniqueness
| |
gactsP(G : {set rT}) : reflect {acts A, on G | to} [acts A, on G | to].
Proof.
apply: (iffP idP) => [nGA x|nGA]; first exact: acts_act.
apply/subsetP=> a Aa /[!inE]; rewrite Aa.
by apply/subsetP=> x; rewrite inE nGA.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
gactsP
| |
gactsM(N1 N2 : {set rT}) :
N1 \subset D -> N2 \subset D ->
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 * N2 | to].
Proof.
move=> sN1D sN2D aAN1 aAN2; apply/gactsP=> x Ax y.
apply/idP/idP; case/mulsgP=> y1 y2 N1y1 N2y2 e.
move: (actKin to Ax y); rewrite e; move<-.
rewrite gactM ?groupV ?(subsetP sN1D y1) ?(subsetP sN2D) //.
by apply: mem_mulg; rewrite ?(gactsP _ aAN1) ?(gactsP _ aAN2) // groupV.
rewrite e gactM // ?(subsetP sN1D y1) ?(subsetP sN2D) //.
by apply: mem_mulg; rewrite ?(gactsP _ aAN1) // ?(gactsP _ aAN2).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
gactsM
| |
gactsI(N1 N2 : {set rT}) :
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 :&: N2 | to].
Proof.
move=> aAN1 aAN2.
apply/subsetP=> x Ax; rewrite !inE Ax /=; apply/subsetP=> y Ny /[1!inE].
case/setIP: Ny=> N1y N2y; rewrite inE ?astabs_act ?N1y ?N2y //.
- by move/subsetP: aAN2; move/(_ x Ax).
- by move/subsetP: aAN1; move/(_ x Ax).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
gactsI
| |
gastabsP(S : {set rT}) (a : aT) :
a \in A -> reflect (forall x, (to x a \in S) = (x \in S)) (a \in 'N(S | to)).
Proof.
move=> Aa; apply: (iffP idP) => [nSa x|nSa]; first exact: astabs_act.
by rewrite !inE Aa; apply/subsetP=> x; rewrite inE nSa.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
gastabsP
| |
qact_dom_doms(H : {group rT}) : H \subset D -> qact_dom to H \subset A.
Proof.
by move=> sHD; apply/subsetP=> x; rewrite qact_domE // inE; case/andP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
qact_dom_doms
| |
acts_qact_doms(H : {group rT}) :
H \subset D -> [acts A, on H | to] -> qact_dom to H :=: A.
Proof.
move=> sHD aH; apply/eqP; rewrite eqEsubset; apply/andP.
split; first exact: qact_dom_doms.
apply/subsetP=> x Ax; rewrite qact_domE //; apply/gastabsP=> //.
by move/gactsP: aH; move/(_ x Ax).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
acts_qact_doms
| |
qacts_cosetpre(H : {group rT}) (K' : {group coset_of H}) :
H \subset D -> [acts A, on H | to] ->
[acts qact_dom to H, on K' | to / H] ->
[acts A, on coset H @*^-1 K' | to].
Proof.
move=> sHD aH aK'; apply/subsetP=> x Ax; move: (Ax) (subsetP aK').
rewrite -{1}(acts_qact_doms sHD aH) => qdx; move/(_ x qdx) => nx.
rewrite !inE Ax; apply/subsetP=> y; case/morphpreP=> Ny /= K'Hy /[1!inE].
apply/morphpreP; split; first by rewrite acts_qact_dom_norm.
by move/gastabsP: nx; move/(_ qdx (coset H y)); rewrite K'Hy qactE.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
qacts_cosetpre
| |
qacts_coset(H K : {group rT}) :
H \subset D -> [acts A, on K | to] ->
[acts qact_dom to H, on (coset H) @* K | to / H].
Proof.
move=> sHD aK.
apply/subsetP=> x qdx; rewrite inE qdx inE; apply/subsetP=> y.
case/morphimP=> z Nz Kz /= e; rewrite e inE qactE // imset_f // inE.
move/gactsP: aK; move/(_ x (subsetP (qact_dom_doms sHD) _ qdx) z); rewrite Kz.
move->; move/acts_act: (acts_qact_dom to H); move/(_ x qdx z).
by rewrite Nz andbT.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
qacts_coset
| |
maxainv(B C : {set rT}) :=
[max C of H |
[&& (H <| B), ~~ (B \subset H) & [acts A, on H | to]]].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
maxainv
| |
maxainv_norm: maxainv K N -> N <| K.
Proof. by move/maxgroupp; case/andP. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
maxainv_norm
| |
maxainv_proper: maxainv K N -> N \proper K.
Proof.
by move/maxgroupp; case/andP; rewrite properE; move/normal_sub->; case/andP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
maxainv_proper
| |
maxainv_sub: maxainv K N -> N \subset K.
Proof. by move=> h; apply: proper_sub; apply: maxainv_proper. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
maxainv_sub
| |
maxainv_ainvar: maxainv K N -> A \subset 'N(N | to).
Proof. by move/maxgroupp; case/and3P. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
maxainv_ainvar
| |
maxainvS: maxainv K N -> N \subset K.
Proof. by move=> pNN; rewrite proper_sub // maxainv_proper. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
maxainvS
| |
maxainv_exists: K :!=: 1 -> {N : {group rT} | maxainv K N}.
Proof.
move=> nt; apply: ex_maxgroup. exists [1 rT]%G.
rewrite /= normal1 subG1 nt /=.
apply/subsetP=> a Da; rewrite !inE Da /= sub1set !inE.
by rewrite /= -actmE // morph1 eqxx.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
maxainv_exists
| |
maxainvM(G H K : {group rT}) :
H \subset D -> K \subset D -> maxainv G H -> maxainv G K ->
H :<>: K -> H * K = G.
Proof.
move: H K => N1 N2 sN1D sN2D pmN1 pmN2 neN12.
have cN12 : commute N1 N2.
apply: normC; apply: (subset_trans (maxainv_sub pmN1)).
by rewrite normal_norm ?maxainv_norm.
wlog nsN21 : G N1 N2 sN1D sN2D pmN1 pmN2 neN12 cN12/ ~~(N1 \subset N2).
move/eqP: (neN12); rewrite eqEsubset negb_and; case/orP=> ns; first by apply.
by rewrite cN12; apply=> //; apply: sym_not_eq.
have nP : N1 * N2 <| G by rewrite normalM ?maxainv_norm.
have sN2P : N2 \subset N1 * N2 by rewrite mulg_subr ?group1.
case/maxgroupP: (pmN1); case/andP=> nN1G pN1G mN1.
case/maxgroupP: (pmN2); case/andP=> nN2G pN2G mN2.
case/andP: pN1G=> nsGN1 ha1; case/andP: pN2G=> nsGN2 ha2.
case e : (G \subset N1 * N2).
by apply/eqP; rewrite eqEsubset e mulG_subG !normal_sub.
have: N1 <*> N2 = N2 by apply: mN2; rewrite /= ?comm_joingE // nP e /= gactsM.
by rewrite comm_joingE // => h; move: nsN21; rewrite -h mulg_subl.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
maxainvM
| |
asimple(K : {set rT}) := maxainv K 1.
Implicit Types (H K : {group rT}) (s : seq {group rT}).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
asimple
| |
asimplePK :
reflect [/\ K :!=: 1
& forall H, H <| K -> [acts A, on H | to] -> H :=: 1 \/ H :=: K]
(asimple K).
Proof.
apply: (iffP idP).
case/maxgroupP; rewrite normal1 /=; case/andP=> nsK1 aK H1.
rewrite eqEsubset negb_and nsK1 /=; split => // H nHK ha.
case eHK : (H :==: K); first by right; apply/eqP.
left; apply: H1; rewrite ?sub1G // nHK; move/negbT: eHK.
by rewrite eqEsubset negb_and normal_sub //=; move->.
case=> ntK h; apply/maxgroupP; split.
move: ntK; rewrite eqEsubset sub1G andbT normal1; move->.
apply/subsetP=> a Da; rewrite !inE Da /= sub1set !inE.
by rewrite /= -actmE // morph1 eqxx.
move=> H /andP[nHK /andP[nsKH ha]] _.
case: (h _ nHK ha)=> // /eqP; rewrite eqEsubset.
by rewrite (negbTE nsKH) andbF.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
asimpleP
| |
acompsK s :=
((last K s) == 1%G) && path [rel x y : {group rT} | maxainv x y] K s.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
acomps
| |
acompsPK s :
reflect (last K s = 1%G /\ path [rel x y : {group rT} | maxainv x y] K s)
(acomps K s).
Proof. by apply: (iffP andP); case; move/eqP. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
acompsP
| |
trivg_acompsK s : acomps K s -> (K :==: 1) = (s == [::]).
Proof.
case/andP=> ls cs; apply/eqP/eqP; last first.
by move=> se; rewrite se /= in ls; apply/eqP.
move=> G1; case: s ls cs => // H s _ /=; case/andP; case/maxgroupP.
by rewrite G1 sub1G andbF.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
trivg_acomps
| |
acomps_consK H s : acomps K (H :: s) -> acomps H s.
Proof. by case/andP => /= ls; case/andP=> _ p; rewrite /acomps ls. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
acomps_cons
| |
asimple_acompsPK s : acomps K s -> reflect (s = [:: 1%G]) (asimple K).
Proof.
move=> cs; apply: (iffP idP); last first.
by move=> se; move: cs; rewrite se /=; case/andP=> /=; rewrite andbT.
case: s cs.
by rewrite /acomps /= andbT; move/eqP->; case/asimpleP; rewrite eqxx.
move=> H s cs sG; apply/eqP.
rewrite eqseq_cons -(trivg_acomps (acomps_cons cs)) andbC andbb.
case/acompsP: cs => /= ls; case/andP=> mH ps.
case/maxgroupP: sG; case/and3P => _ ntG _ ->; rewrite ?sub1G //.
rewrite (maxainv_norm mH); case/andP: (maxainv_proper mH)=> _ ->.
exact: (maxainv_ainvar mH).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
asimple_acompsP
| |
exists_acompsK : exists s, acomps K s.
Proof.
elim: {K} #|K| {1 3}K (leqnn #|K|) => [K | n Hi K cK].
by rewrite leqNgt cardG_gt0.
case/orP: (orbN (asimple K)) => [sK | nsK].
by exists [:: (1%G : {group rT})]; rewrite /acomps eqxx /= andbT.
case/orP: (orbN (K :==: 1))=> [tK | ntK].
by exists (Nil _); rewrite /acomps /= andbT.
case: (maxainv_exists ntK)=> N pmN.
have cN: #|N| <= n.
by rewrite -ltnS (leq_trans _ cK) // proper_card // (maxainv_proper pmN).
case: (Hi _ cN)=> s; case/andP=> lasts ps; exists [:: N & s]; rewrite /acomps.
by rewrite last_cons lasts /= pmN.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
exists_acomps
| |
maxainv_asimple_quo(G H : {group rT}) :
H \subset D -> maxainv to G H -> asimple (to / H) (G / H).
Proof.
move=> sHD /maxgroupP[/and3P[nHG pHG aH] Hmax].
apply/asimpleP; split; first by rewrite -subG1 quotient_sub1 ?normal_norm.
move=> K' nK'Q aK'.
have: (K' \proper (G / H)) || (G / H == K').
by rewrite properE eqEsubset andbC (normal_sub nK'Q) !andbT orbC orbN.
case/orP=> [ pHQ | eQH]; last by right; apply sym_eq; apply/eqP.
left; pose K := ((coset H) @*^-1 K')%G.
have eK'I : K' \subset (coset H) @* 'N(H).
by rewrite (subset_trans (normal_sub nK'Q)) ?morphimS ?normal_norm.
have eKK' : K' :=: K / H by rewrite /(K / H) morphpreK //=.
suff eKH : K :=: H by rewrite -trivg_quotient eKK' eKH.
have sHK : H \subset K by rewrite -ker_coset kerE morphpreS // sub1set group1.
apply: Hmax => //; apply/and3P; split; last exact: qacts_cosetpre.
by rewrite -(quotientGK nHG) cosetpre_normal.
by move: (proper_subn pHQ); rewrite sub_morphim_pre ?normal_norm.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
maxainv_asimple_quo
| |
asimple_quo_maxainv(G H : {group rT}) :
H \subset D -> G \subset D -> [acts A, on G | to] -> [acts A, on H | to] ->
H <| G -> asimple (to / H) (G / H) ->
maxainv to G H.
Proof.
move=> sHD sGD aG aH nHG /asimpleP[ntQ maxQ]; apply/maxgroupP; split.
by rewrite nHG -quotient_sub1 ?normal_norm // subG1 ntQ.
move=> K /and3P[nKG nsGK aK] sHK.
pose K' := (K / H)%G.
have K'dQ : K' <| (G / H)%G by apply: morphim_normal.
have nKH : H <| K by rewrite (normalS _ _ nHG) // normal_sub.
have: K' :=: 1%G \/ K' :=: (G / H).
apply: (maxQ K' K'dQ) => /=.
apply/subsetP=> x Adx. rewrite inE Adx /= inE. apply/subsetP=> y.
rewrite quotientE; case/morphimP=> z Nz Kz ->; rewrite /= !inE qactE //.
have ntoyx : to z x \in 'N(H) by rewrite (acts_qact_dom_norm Adx).
apply/morphimP; exists (to z x) => //.
suff h: qact_dom to H \subset A.
by rewrite astabs_act // (subsetP aK) //; apply: (subsetP h).
by apply/subsetP=> t; rewrite qact_domE // inE; case/andP.
case=> [|/quotient_injG /[!inE]/(_ nKH nHG) c]; last by rewrite c subxx in nsGK.
rewrite /= -trivg_quotient => tK'; apply: (congr1 (@gval _)); move: tK'.
by apply: (@quotient_injG _ H); rewrite ?inE /= ?normal_refl.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
asimple_quo_maxainv
| |
asimpleI(N1 N2 : {group rT}) :
N2 \subset 'N(N1) -> N1 \subset D ->
[acts A, on N1 | to] -> [acts A, on N2 | to] ->
asimple (to / N1) (N2 / N1) ->
asimple (to / (N2 :&: N1)) (N2 / (N2 :&: N1)).
Proof.
move=> nN21 sN1D aN1 aN2 /asimpleP[ntQ1 max1].
have [f1 [f1e f1ker f1pre f1im]] := restrmP (coset_morphism N1) nN21.
have hf2' : N2 \subset 'N(N2 :&: N1) by apply: normsI => //; rewrite normG.
have hf2'' : 'ker (coset (N2 :&: N1)) \subset 'ker f1.
by rewrite f1ker !ker_coset.
pose f2 := factm_morphism hf2'' hf2'.
apply/asimpleP; split.
rewrite /= setIC; apply/negP; move: (second_isog nN21); move/isog_eq1->.
by apply/negP.
move=> H nHQ2 aH; pose K := f2 @* H.
have nKQ1 : K <| N2 / N1.
rewrite (_ : N2 / N1 = f2 @* (N2 / (N2 :&: N1))) ?morphim_normal //.
by rewrite morphim_factm f1im.
have sqA : qact_dom to N1 \subset A.
by apply/subsetP=> t; rewrite qact_domE // inE; case/andP.
have nNN2 : (N2 :&: N1) <| N2.
by rewrite /normal subsetIl; apply: normsI => //; apply: normG.
have aKQ1 : [acts qact_dom to N1, on K | to / N1].
pose H':= coset (N2 :&: N1)@*^-1 H.
have eHH' : H :=: H' / (N2 :&: N1) by rewrite cosetpreK.
have -> : K :=: f1 @* H' by rewrite /K eHH' morphim_factm.
have sH'N2 : H' \subset N2.
rewrite /H' eHH' quotientGK ?normal_cosetpre //=.
by rewrite sub_cosetpre_quo ?normal_sub.
have -> : f1 @* H' = coset N1 @* H' by rewrite f1im //=.
apply: qacts_coset => //; apply: qacts_cosetpre => //; last exact: gactsI.
by apply: (subset_tr
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
asimpleI
| |
StrongJordanHolderUniqueness(G : {group rT}) (s1 s2 : seq {group rT}) :
G \subset D -> acomps to G s1 -> acomps to G s2 ->
perm_eq (mkfactors G s1) (mkfactors G s2).
Proof.
have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => cG hsD cs1 cs2.
case/orP: (orbN (G :==: 1)) => [tG | ntG].
have -> : s1 = [::] by apply/eqP; rewrite -(trivg_acomps cs1).
have -> : s2 = [::] by apply/eqP; rewrite -(trivg_acomps cs2).
by rewrite /= perm_refl.
case/orP: (orbN (asimple to G))=> [sG | nsG].
have -> : s1 = [:: 1%G ] by apply/(asimple_acompsP cs1).
have -> : s2 = [:: 1%G ] by apply/(asimple_acompsP cs2).
by rewrite /= perm_refl.
case es1: s1 cs1 => [|N1 st1] cs1.
by move: (trivg_comps cs1); rewrite eqxx; move/negP:ntG.
case es2: s2 cs2 => [|N2 st2] cs2 {s1 es1}.
by move: (trivg_comps cs2); rewrite eqxx; move/negP:ntG.
case/andP: cs1 => /= lst1; case/andP=> maxN_1 pst1.
case/andP: cs2 => /= lst2; case/andP=> maxN_2 pst2.
have sN1D : N1 \subset D.
by apply: subset_trans hsD; apply: maxainv_sub maxN_1.
have sN2D : N2 \subset D.
by apply: subset_trans hsD; apply: maxainv_sub maxN_2.
have cN1 : #|N1| < n.
by rewrite -ltnS (leq_trans _ cG) ?ltnS ?proper_card ?(maxainv_proper maxN_1).
have cN2 : #|N2| < n.
by rewrite -ltnS (leq_trans _ cG) ?ltnS ?proper_card ?(maxainv_proper maxN_2).
case: (N1 =P N2) {s2 es2} => [eN12 |].
by rewrite eN12 /= perm_cons Hi // /acomps ?lst2 //= -eN12 lst1.
move/eqP; rewrite -val_eqE /=; move/eqP=> neN12.
have nN1G : N1 <| G by apply: (maxainv_norm maxN_1).
h
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] |
solvable/jordanholder.v
|
StrongJordanHolderUniqueness
| |
charsimpleA := [min A of G | G :!=: 1 & G \char A].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
charsimple
| |
FrattiniA := \bigcap_(G : {group gT} | maximal_eq G A) G.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Frattini
| |
Frattini_groupA : {group gT} := Eval hnf in [group of Frattini A].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Frattini_group
| |
FittingA := \big[dprod/1]_(p <- primes #|A|) 'O_p(A).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Fitting
| |
Fitting_group_setG : group_set (Fitting G).
Proof.
suffices [F ->]: exists F : {group gT}, Fitting G = F by apply: groupP.
rewrite /Fitting; elim: primes (primes_uniq #|G|) => [_|p r IHr] /=.
by exists [1 gT]%G; rewrite big_nil.
case/andP=> rp /IHr[F defF]; rewrite big_cons defF.
suffices{IHr} /and3P[p'F sFG nFG]: p^'.-group F && (F <| G).
have nFGp: 'O_p(G) \subset 'N(F) := gFsub_trans _ nFG.
have pGp: p.-group('O_p(G)) := pcore_pgroup p G.
have{pGp} tiGpF: 'O_p(G) :&: F = 1 by rewrite coprime_TIg ?(pnat_coprime pGp).
exists ('O_p(G) <*> F)%G; rewrite dprodEY // (sameP commG1P trivgP) -tiGpF.
by rewrite subsetI commg_subl commg_subr (subset_trans sFG) // gFnorm.
move/bigdprodWY: defF => <- {F}; elim: r rp => [_|q r IHr] /=.
by rewrite big_nil gen0 pgroup1 normal1.
rewrite inE eq_sym big_cons -joingE -joing_idr => /norP[qp /IHr {IHr}].
set F := <<_>> => /andP[p'F nsFG].
rewrite norm_joinEl /= -/F; last exact/gFsub_trans/normal_norm.
by rewrite pgroupM p'F normalM ?pcore_normal //= (pi_pgroup (pcore_pgroup q G)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Fitting_group_set
| |
Fitting_groupG := group (Fitting_group_set G).
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Fitting_group
| |
criticalA B :=
[/\ A \char B,
Frattini A \subset 'Z(A),
[~: B, A] \subset 'Z(A)
& 'C_B(A) = 'Z(A)].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
critical
| |
specialA := Frattini A = 'Z(A) /\ A^`(1) = 'Z(A).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
special
| |
extraspecialA := special A /\ prime #|'Z(A)|.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
extraspecial
| |
SCNB := [set A : {group gT} | A <| B & 'C_B(A) == A].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
SCN
| |
SCN_atn B := [set A in SCN B | n <= 'r(A)].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
SCN_at
| |
p_maximal_normal: maximal M P -> M <| P.
Proof.
case/maxgroupP=> /andP[sMP sPM] maxM; rewrite /normal sMP.
have:= subsetIl P 'N(M); rewrite subEproper.
case/predU1P=> [/setIidPl-> // | /maxM/= SNM]; case/negP: sPM.
rewrite (nilpotent_sub_norm (pgroup_nil pP) sMP) //.
by rewrite SNM // subsetI sMP normG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
p_maximal_normal
| |
p_maximal_index: maximal M P -> #|P : M| = p.
Proof.
move=> maxM; have nM := p_maximal_normal maxM.
rewrite -card_quotient ?normal_norm //.
rewrite -(quotient_maximal _ nM) ?normal_refl // trivg_quotient in maxM.
case/maxgroupP: maxM; rewrite properEneq eq_sym sub1G andbT /=.
case/(pgroup_pdiv (quotient_pgroup M pP)) => p_pr /Cauchy[] // xq.
rewrite /order -cycle_subG subEproper => /predU1P[-> // | sxPq oxq_p _].
by move/(_ _ sxPq (sub1G _)) => xq1; rewrite -oxq_p xq1 cards1 in p_pr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
p_maximal_index
| |
p_index_maximal: M \subset P -> prime #|P : M| -> maximal M P.
Proof.
move=> sMP /primeP[lt1PM pr_PM].
apply/maxgroupP; rewrite properEcard sMP -(Lagrange sMP).
rewrite -{1}(muln1 #|M|) ltn_pmul2l //; split=> // H sHP sMH.
apply/eqP; rewrite eq_sym eqEcard sMH.
case/orP: (pr_PM _ (indexSg sMH (proper_sub sHP))) => /eqP iM.
by rewrite -(Lagrange sMH) iM muln1 /=.
by have:= proper_card sHP; rewrite -(Lagrange sMH) iM Lagrange ?ltnn.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
p_index_maximal
| |
Phi_subG : 'Phi(G) \subset G.
Proof. by rewrite bigcap_inf // /maximal_eq eqxx. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_sub
| |
Phi_sub_maxG M : maximal M G -> 'Phi(G) \subset M.
Proof. by move=> maxM; rewrite bigcap_inf // /maximal_eq predU1r. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_sub_max
| |
Phi_properG : G :!=: 1 -> 'Phi(G) \proper G.
Proof.
move/eqP; case/maximal_exists: (sub1G G) => [<- //| [M maxM _] _].
exact: sub_proper_trans (Phi_sub_max maxM) (maxgroupp maxM).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_proper
| |
Phi_nongenG X : 'Phi(G) <*> X = G -> <<X>> = G.
Proof.
move=> defG; have: <<X>> \subset G by rewrite -{1}defG genS ?subsetUr.
case/maximal_exists=> //= [[M maxM]]; rewrite gen_subG => sXM.
case/andP: (maxgroupp maxM) => _ /negP[].
by rewrite -defG gen_subG subUset Phi_sub_max.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_nongen
| |
Frattini_continuous(rT : finGroupType) G (f : {morphism G >-> rT}) :
f @* 'Phi(G) \subset 'Phi(f @* G).
Proof.
apply/bigcapsP=> M maxM; rewrite sub_morphim_pre ?Phi_sub // bigcap_inf //.
have {2}<-: f @*^-1 (f @* G) = G by rewrite morphimGK ?subsetIl.
by rewrite morphpre_maximal_eq ?maxM //; case/maximal_eqP: maxM.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Frattini_continuous
| |
Frattini_igFun:= [igFun by Phi_sub & Frattini_continuous].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Frattini_igFun
| |
Frattini_gFun:= [gFun by Frattini_continuous].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Frattini_gFun
| |
Phi_charG : 'Phi(G) \char G.
Proof. exact: gFchar. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_char
| |
Phi_normalG : 'Phi(G) <| G.
Proof. exact: gFnormal. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_normal
| |
injm_PhirT D G (f : {morphism D >-> rT}) :
'injm f -> G \subset D -> f @* 'Phi(G) = 'Phi(f @* G).
Proof. exact: injmF. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
injm_Phi
| |
isog_PhirT G (H : {group rT}) : G \isog H -> 'Phi(G) \isog 'Phi(H).
Proof. exact: gFisog. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
isog_Phi
| |
PhiJG x : 'Phi(G :^ x) = 'Phi(G) :^ x.
Proof.
rewrite -{1}(setIid G) -(setIidPr (Phi_sub G)) -!morphim_conj.
by rewrite injm_Phi ?injm_conj.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
PhiJ
| |
Phi_quotient_idG : 'Phi (G / 'Phi(G)) = 1.
Proof.
apply/trivgP; rewrite -cosetpreSK cosetpre1 /=; apply/bigcapsP=> M maxM.
have nPhi := Phi_normal G; have nPhiM: 'Phi(G) <| M.
by apply: normalS nPhi; [apply: bigcap_inf | case/maximal_eqP: maxM].
by rewrite sub_cosetpre_quo ?bigcap_inf // quotient_maximal_eq.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_quotient_id
| |
Phi_quotient_cyclicG : cyclic (G / 'Phi(G)) -> cyclic G.
Proof.
case/cyclicP=> /= Px; case: (cosetP Px) => x nPx ->{Px} defG.
apply/cyclicP; exists x; symmetry; apply: Phi_nongen.
rewrite -joing_idr norm_joinEr -?quotientK ?cycle_subG //.
by rewrite /quotient morphim_cycle //= -defG quotientGK ?Phi_normal.
Qed.
Variables (p : nat) (P : {group gT}).
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_quotient_cyclic
| |
trivg_Phi: p.-group P -> ('Phi(P) == 1) = p.-abelem P.
Proof.
move=> pP; case: (eqsVneq P 1) => [P1 | ntP].
by rewrite P1 abelem1 -subG1 -P1 Phi_sub.
have [p_pr _ _] := pgroup_pdiv pP ntP.
apply/eqP/idP=> [trPhi | abP].
apply/abelemP=> //; split=> [|x Px].
apply/commG1P/trivgP; rewrite -trPhi.
apply/bigcapsP=> M /predU1P[-> | maxM]; first exact: der1_subG.
have /andP[_ nMP]: M <| P := p_maximal_normal pP maxM.
rewrite der1_min // cyclic_abelian // prime_cyclic // card_quotient //.
by rewrite (p_maximal_index pP).
apply/set1gP; rewrite -trPhi; apply/bigcapP=> M.
case/predU1P=> [-> | maxM]; first exact: groupX.
have /andP[_ nMP] := p_maximal_normal pP maxM.
have nMx : x \in 'N(M) by apply: subsetP Px.
apply: coset_idr; rewrite ?groupX ?morphX //=; apply/eqP.
rewrite -(p_maximal_index pP maxM) -card_quotient // -order_dvdn cardSg //=.
by rewrite cycle_subG mem_quotient.
apply/trivgP/subsetP=> x Phi_x; rewrite -cycle_subG.
have Px: x \in P by apply: (subsetP (Phi_sub P)).
have sxP: <[x]> \subset P by rewrite cycle_subG.
case/splitsP: (abelem_splits abP sxP) => K /complP[tiKx defP].
have [-> | nt_x] := eqVneq x 1; first by rewrite cycle1.
have oxp := abelem_order_p abP Px nt_x.
rewrite /= -tiKx subsetI subxx cycle_subG.
apply: (bigcapP Phi_x); apply/orP; right.
apply: p_index_maximal; rewrite -?divgS -defP ?mulG_subr //.
by rewrite (TI_cardMg tiKx) mulnK // [#|_|]oxp.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
trivg_Phi
| |
Phi_quotient_abelem: p.-abelem (P / 'Phi(P)).
Proof. by rewrite -trivg_Phi ?morphim_pgroup //= Phi_quotient_id. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_quotient_abelem
| |
Phi_joing: 'Phi(P) = P^`(1) <*> 'Mho^1(P).
Proof.
have [sPhiP nPhiP] := andP (Phi_normal P).
apply/eqP; rewrite eqEsubset join_subG.
case: (eqsVneq P 1) => [-> | ntP] in sPhiP *.
by rewrite /= (trivgP sPhiP) sub1G der_subS Mho_sub.
have [p_pr _ _] := pgroup_pdiv pP ntP.
have [abP x1P] := abelemP p_pr Phi_quotient_abelem.
apply/andP; split.
have nMP: P \subset 'N(P^`(1) <*> 'Mho^1(P)) by rewrite normsY // !gFnorm.
rewrite -quotient_sub1 ?gFsub_trans //=.
suffices <-: 'Phi(P / (P^`(1) <*> 'Mho^1(P))) = 1 by apply: morphimF.
apply/eqP; rewrite (trivg_Phi (morphim_pgroup _ pP)) /= -quotientE.
apply/abelemP=> //; rewrite [abelian _]quotient_cents2 ?joing_subl //.
split=> // _ /morphimP[x Nx Px ->] /=.
rewrite -morphX //= coset_id // (MhoE 1 pP) joing_idr expn1.
by rewrite mem_gen //; apply/setUP; right; apply: imset_f.
rewrite -quotient_cents2 // [_ \subset 'C(_)]abP (MhoE 1 pP) gen_subG /=.
apply/subsetP=> _ /imsetP[x Px ->]; rewrite expn1.
have nPhi_x: x \in 'N('Phi(P)) by apply: (subsetP nPhiP).
by rewrite coset_idr ?groupX ?morphX ?x1P ?mem_morphim.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_joing
| |
Phi_Mho: abelian P -> 'Phi(P) = 'Mho^1(P).
Proof. by move=> cPP; rewrite Phi_joing (derG1P cPP) joing1G. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_Mho
| |
PhiSG H : p.-group H -> G \subset H -> 'Phi(G) \subset 'Phi(H).
Proof.
move=> pH sGH; rewrite (Phi_joing pH) (Phi_joing (pgroupS sGH pH)).
by rewrite genS // setUSS ?dergS ?MhoS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
PhiS
| |
morphim_PhirT P D (f : {morphism D >-> rT}) :
p.-group P -> P \subset D -> f @* 'Phi(P) = 'Phi(f @* P).
Proof.
move=> pP sPD; rewrite !(@Phi_joing _ p) ?morphim_pgroup //.
rewrite morphim_gen ?subUset ?gFsub_trans // morphimU -joingE.
by rewrite morphimR ?morphim_Mho.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
morphim_Phi
| |
quotient_PhiP H :
p.-group P -> P \subset 'N(H) -> 'Phi(P) / H = 'Phi(P / H).
Proof. exact: morphim_Phi. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
quotient_Phi
| |
Phi_minG H :
p.-group G -> G \subset 'N(H) -> p.-abelem (G / H) -> 'Phi(G) \subset H.
Proof.
move=> pG nHG; rewrite -trivg_Phi ?quotient_pgroup // -subG1 /=.
by rewrite -(quotient_Phi pG) ?quotient_sub1 // gFsub_trans.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_min
| |
Phi_cprodG H K :
p.-group G -> H \* K = G -> 'Phi(H) \* 'Phi(K) = 'Phi(G).
Proof.
move=> pG defG; have [_ /mulG_sub[sHG sKG] cHK] := cprodP defG.
rewrite cprodEY /=; last by rewrite (centSS (Phi_sub _) (Phi_sub _)).
rewrite !(Phi_joing (pgroupS _ pG)) //=.
have /cprodP[_ <- /cent_joinEr <-] := der_cprod 1 defG.
have /cprodP[_ <- /cent_joinEr <-] := Mho_cprod 1 defG.
by rewrite !joingA /= -!(joingA H^`(1)) (joingC K^`(1)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_cprod
| |
Phi_mulgH K :
p.-group H -> p.-group K -> K \subset 'C(H) ->
'Phi(H * K) = 'Phi(H) * 'Phi(K).
Proof.
move=> pH pK cHK; have defHK := cprodEY cHK.
have [|_ ->] /= := cprodP (Phi_cprod _ defHK); rewrite cent_joinEr //.
by rewrite pgroupM pH.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg ",
"From mathcomp Require Import countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule"
] |
solvable/maximal.v
|
Phi_mulg
|
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