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 |
|---|---|---|---|---|---|---|
charsimplePG :
reflect (G :!=: 1 /\ forall K, K :!=: 1 -> K \char G -> K :=: G)
(charsimple G).
Proof.
apply: (iffP mingroupP); rewrite char_refl andbT => -[ntG simG].
by split=> // K ntK chK; apply: simG; rewrite ?ntK // char_sub.
by split=> // K /andP[ntK chK] _; apply: simG.
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
|
charsimpleP
| |
Fitting_normalG : 'F(G) <| G.
Proof.
rewrite -['F(G)](bigdprodWY (erefl 'F(G))).
elim/big_rec: _ => [|p H _ nsHG]; first by rewrite gen0 normal1.
by rewrite -[<<_>>]joing_idr normalY ?pcore_normal.
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_normal
| |
Fitting_subG : 'F(G) \subset G.
Proof. by rewrite normal_sub ?Fitting_normal. 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_sub
| |
Fitting_nilG : nilpotent 'F(G).
Proof.
apply: (bigdprod_nil (erefl 'F(G))) => p _.
exact: pgroup_nil (pcore_pgroup p 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_nil
| |
Fitting_maxG H : H <| G -> nilpotent H -> H \subset 'F(G).
Proof.
move=> nsHG nilH; rewrite -(Sylow_gen H) gen_subG.
apply/bigcupsP=> P /SylowP[p _ sylP].
case Gp: (p \in \pi(G)); last first.
rewrite card1_trivg ?sub1G // (card_Hall sylP).
rewrite part_p'nat // (pnat_dvd (cardSg (normal_sub nsHG))) //.
by rewrite /pnat cardG_gt0 all_predC has_pred1 Gp.
rewrite {P sylP}(nilpotent_Hall_pcore nilH sylP).
rewrite -(bigdprodWY (erefl 'F(G))) sub_gen //.
rewrite -(filter_pi_of (ltnSn _)) big_filter big_mkord.
apply: (bigcup_max (Sub p _)) => //= [|_].
by have:= Gp; rewrite ltnS mem_primes => /and3P[_ ntG /dvdn_leq->].
by rewrite pcore_max ?pcore_pgroup ?gFnormal_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
|
Fitting_max
| |
pcore_Fittingpi G : 'O_pi('F(G)) \subset 'O_pi(G).
Proof. by rewrite pcore_max ?pcore_pgroup ?gFnormal_trans ?Fitting_normal. 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
|
pcore_Fitting
| |
p_core_Fittingp G : 'O_p('F(G)) = 'O_p(G).
Proof.
apply/eqP; rewrite eqEsubset pcore_Fitting pcore_max ?pcore_pgroup //.
apply: normalS (normal_sub (Fitting_normal _)) (pcore_normal _ _).
exact: Fitting_max (pcore_normal _ _) (pgroup_nil (pcore_pgroup _ _)).
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_core_Fitting
| |
nilpotent_FittingG : nilpotent G -> 'F(G) = G.
Proof.
by move=> nilG; apply/eqP; rewrite eqEsubset Fitting_sub Fitting_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
|
nilpotent_Fitting
| |
Fitting_eq_pcorep G : 'O_p^'(G) = 1 -> 'F(G) = 'O_p(G).
Proof.
move=> p'G1; have /dprodP[_ /= <- _ _] := nilpotent_pcoreC p (Fitting_nil G).
by rewrite p_core_Fitting ['O_p^'(_)](trivgP _) ?mulg1 // -p'G1 pcore_Fitting.
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_eq_pcore
| |
FittingEgenG :
'F(G) = <<\bigcup_(p < #|G|.+1 | (p : nat) \in \pi(G)) 'O_p(G)>>.
Proof.
apply/eqP; rewrite eqEsubset gen_subG /=.
rewrite -{1}(bigdprodWY (erefl 'F(G))) (big_nth 0) big_mkord genS.
by apply/bigcupsP=> p _; rewrite -p_core_Fitting pcore_sub.
apply/bigcupsP=> [[i /= lti]] _; set p := nth _ _ i.
have pi_p: p \in \pi(G) by rewrite mem_nth.
have p_dv_G: p %| #|G| by rewrite mem_primes in pi_p; case/and3P: pi_p.
have lepG: p < #|G|.+1 by rewrite ltnS dvdn_leq.
by rewrite (bigcup_max (Ordinal lepG)).
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
|
FittingEgen
| |
morphim_Fitting: GFunctor.pcontinuous (@Fitting).
Proof.
move=> gT rT G D f; apply: Fitting_max.
by rewrite morphim_normal ?Fitting_normal.
by rewrite morphim_nil ?Fitting_nil.
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_Fitting
| |
FittingSgT (G H : {group gT}) : H \subset G -> H :&: 'F(G) \subset 'F(H).
Proof.
move=> sHG; rewrite -{2}(setIidPl sHG).
do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; apply: morphim_Fitting.
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
|
FittingS
| |
FittingJgT (G : {group gT}) x : 'F(G :^ x) = 'F(G) :^ x.
Proof.
rewrite !FittingEgen -genJ /= cardJg; symmetry; congr <<_>>.
rewrite (big_morph (conjugate^~ x) (fun A B => conjUg A B x) (imset0 _)).
by apply: eq_bigr => p _; rewrite pcoreJ.
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
|
FittingJ
| |
Fitting_igFun:= [igFun by Fitting_sub & morphim_Fitting].
|
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_igFun
| |
Fitting_gFun:= [gFun by morphim_Fitting].
|
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_gFun
| |
Fitting_pgFun:= [pgFun by morphim_Fitting].
|
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_pgFun
| |
Fitting_char: 'F(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
|
Fitting_char
| |
injm_Fitting: 'injm f -> G \subset D -> f @* 'F(G) = 'F(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_Fitting
| |
isog_Fitting(H : {group rT}) : G \isog H -> 'F(G) \isog 'F(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_Fitting
| |
minnormal_charsimpleG H : minnormal H G -> charsimple H.
Proof.
case/mingroupP=> /andP[ntH nHG] minH.
apply/charsimpleP; split=> // K ntK chK.
by apply: minH; rewrite ?ntK (char_sub chK, char_norm_trans chK).
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
|
minnormal_charsimple
| |
maxnormal_charsimpleG H L :
G <| L -> maxnormal H G L -> charsimple (G / H).
Proof.
case/andP=> sGL nGL /maxgroupP[/andP[/andP[sHG not_sGH] nHL] maxH].
have nHG: G \subset 'N(H) := subset_trans sGL nHL.
apply/charsimpleP; rewrite -subG1 quotient_sub1 //; split=> // HK ntHK chHK.
case/(inv_quotientN _): (char_normal chHK) => [|K defHK sHK]; first exact/andP.
case/andP; rewrite subEproper defHK => /predU1P[-> // | ltKG] nKG.
have nHK: H <| K by rewrite /normal sHK (subset_trans (proper_sub ltKG)).
case/negP: ntHK; rewrite defHK -subG1 quotient_sub1 ?normal_norm //.
rewrite (maxH K) // ltKG -(quotientGK nHK) -defHK norm_quotient_pre //.
by rewrite (char_norm_trans chHK) ?quotient_norms.
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
|
maxnormal_charsimple
| |
abelem_split_dprodrT p (A B : {group rT}) :
p.-abelem A -> B \subset A -> exists C : {group rT}, B \x C = A.
Proof.
move=> abelA sBA; have [_ cAA _]:= and3P abelA.
case/splitsP: (abelem_splits abelA sBA) => C /complP[tiBC defA].
by exists C; rewrite dprodE // (centSS _ sBA cAA) // -defA mulG_subr.
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
|
abelem_split_dprod
| |
p_abelem_split1rT p (A : {group rT}) x :
p.-abelem A -> x \in A ->
exists B : {group rT}, [/\ B \subset A, #|B| = #|A| %/ #[x] & <[x]> \x B = A].
Proof.
move=> abelA Ax; have sxA: <[x]> \subset A by rewrite cycle_subG.
have [B defA] := abelem_split_dprod abelA sxA.
have [_ defxB _ ti_xB] := dprodP defA.
have sBA: B \subset A by rewrite -defxB mulG_subr.
by exists B; split; rewrite // -defxB (TI_cardMg ti_xB) mulKn ?order_gt0.
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_abelem_split1
| |
abelem_charsimplep G : p.-abelem G -> G :!=: 1 -> charsimple G.
Proof.
move=> abelG ntG; apply/charsimpleP; split=> // K ntK /charP[sKG chK].
case/eqVproper: sKG => // /properP[sKG [x Gx notKx]].
have ox := abelem_order_p abelG Gx (group1_contra notKx).
have [A [sAG oA defA]] := p_abelem_split1 abelG Gx.
case/trivgPn: ntK => y Ky nty; have Gy := subsetP sKG y Ky.
have{nty} oy := abelem_order_p abelG Gy nty.
have [B [sBG oB defB]] := p_abelem_split1 abelG Gy.
have: isog A B; last case/isogP=> fAB injAB defAB.
rewrite (isog_abelem_card _ (abelemS sAG abelG)) (abelemS sBG) //=.
by rewrite oA oB ox oy.
have: isog <[x]> <[y]>; last case/isogP=> fxy injxy /= defxy.
by rewrite isog_cyclic_card ?cycle_cyclic // [#|_|]oy -ox eqxx.
have cfxA: fAB @* A \subset 'C(fxy @* <[x]>).
by rewrite defAB defxy; case/dprodP: defB.
have injf: 'injm (dprodm defA cfxA).
by rewrite injm_dprodm injAB injxy defAB defxy; apply/eqP; case/dprodP: defB.
case/negP: notKx; rewrite -cycle_subG -(injmSK injf) ?cycle_subG //=.
rewrite morphim_dprodml // defxy cycle_subG /= chK //.
have [_ {4}<- _ _] := dprodP defB; have [_ {3}<- _ _] := dprodP defA.
by rewrite morphim_dprodm // defAB defxy.
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
|
abelem_charsimple
| |
charsimple_dprodG : charsimple G ->
exists H : {group gT}, [/\ H \subset G, simple H
& exists2 I : {set {perm gT}}, I \subset Aut G
& \big[dprod/1]_(f in I) f @: H = G].
Proof.
case/charsimpleP=> ntG simG.
have [H minH sHG]: {H : {group gT} | minnormal H G & H \subset G}.
by apply: mingroup_exists; rewrite ntG normG.
case/mingroupP: minH => /andP[ntH nHG] minH.
pose Iok (I : {set {perm gT}}) :=
(I \subset Aut G) &&
[exists (M : {group gT} | M <| G), \big[dprod/1]_(f in I) f @: H == M].
have defH: (1 : {perm gT}) @: H = H.
apply/eqP; rewrite eqEcard card_imset ?leqnn; last exact: perm_inj.
by rewrite andbT; apply/subsetP=> _ /imsetP[x Hx ->]; rewrite perm1.
have [|I] := @maxset_exists _ Iok 1.
rewrite /Iok sub1G; apply/existsP; exists H.
by rewrite /normal sHG nHG (big_pred1 1) => [|f]; rewrite ?defH /= ?inE.
case/maxsetP=> /andP[Aut_I /exists_eq_inP[M /andP[sMG nMG] defM]] maxI.
rewrite sub1set=> ntI; case/eqVproper: sMG => [defG | /andP[sMG not_sGM]].
exists H; split=> //; last by exists I; rewrite ?defM.
apply/mingroupP; rewrite ntH normG; split=> // N /andP[ntN nNH] sNH.
apply: minH => //; rewrite ntN /= -defG.
move: defM; rewrite (bigD1 1) //= defH; case/dprodP=> [[_ K _ ->] <- cHK _].
by rewrite mul_subG // cents_norm // (subset_trans cHK) ?centS.
have defG: <<\bigcup_(f in Aut G) f @: H>> = G.
have sXG: \bigcup_(f in Aut G) f @: H \subset G.
by apply/bigcupsP=> f Af; rewrite -(im_autm Af) morphimEdom imse
...
|
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
|
charsimple_dprod
| |
simple_sol_primeG : solvable G -> simple G -> prime #|G|.
Proof.
move=> solG /simpleP[ntG simG].
have{solG} cGG: abelian G.
apply/commG1P; case/simG: (der_normal 1 G) => // /eqP/idPn[].
by rewrite proper_neq // (sol_der1_proper solG).
case: (trivgVpdiv G) ntG => [-> | [p p_pr]]; first by rewrite eqxx.
case/Cauchy=> // x Gx oxp _; move: p_pr; rewrite -oxp orderE.
have: <[x]> <| G by rewrite -sub_abelian_normal ?cycle_subG.
by case/simG=> -> //; rewrite cards1.
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
|
simple_sol_prime
| |
charsimple_solvableG : charsimple G -> solvable G -> is_abelem G.
Proof.
case/charsimple_dprod=> H [sHG simH [I Aut_I defG]] solG.
have p_pr: prime #|H| by apply: simple_sol_prime (solvableS sHG solG) simH.
set p := #|H| in p_pr; apply/is_abelemP; exists p => //.
elim/big_rec: _ (G) defG => [_ <-|f B If IH_B M defM]; first exact: abelem1.
have [Af [[_ K _ defB] _ _ _]] := (subsetP Aut_I f If, dprodP defM).
rewrite (dprod_abelem p defM) defB IH_B // andbT -(autmE Af) -morphimEsub //=.
rewrite morphim_abelem ?abelemE // exponent_dvdn.
by rewrite cyclic_abelian ?prime_cyclic.
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
|
charsimple_solvable
| |
minnormal_solvableL G H :
minnormal H L -> H \subset G -> solvable G ->
[/\ L \subset 'N(H), H :!=: 1 & is_abelem H].
Proof.
move=> minH sHG solG; have /andP[ntH nHL] := mingroupp minH.
split=> //; apply: (charsimple_solvable (minnormal_charsimple minH)).
exact: solvableS solG.
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
|
minnormal_solvable
| |
solvable_norm_abelemL G :
solvable G -> G <| L -> G :!=: 1 ->
exists H : {group gT}, [/\ H \subset G, H <| L, H :!=: 1 & is_abelem H].
Proof.
move=> solG /andP[sGL nGL] ntG.
have [H minH sHG]: {H : {group gT} | minnormal H L & H \subset G}.
by apply: mingroup_exists; rewrite ntG.
have [nHL ntH abH] := minnormal_solvable minH sHG solG.
by exists H; split; rewrite // /normal (subset_trans sHG).
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
|
solvable_norm_abelem
| |
trivg_FittingG : solvable G -> ('F(G) == 1) = (G :==: 1).
Proof.
move=> solG; apply/idP/idP=> [F1 | /eqP->]; last by rewrite gF1.
apply/idPn=> /(solvable_norm_abelem solG (normal_refl _))[M [_ nsMG ntM]].
case/is_abelemP=> p _ /and3P[pM _ _]; case/negP: ntM.
by rewrite -subG1 -(eqP F1) Fitting_max ?(pgroup_nil pM).
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_Fitting
| |
Fitting_pcorepi G : 'F('O_pi(G)) = 'O_pi('F(G)).
Proof.
apply/eqP; rewrite eqEsubset.
rewrite (subset_trans _ (pcoreS _ (Fitting_sub _))); last first.
by rewrite subsetI Fitting_sub Fitting_max ?Fitting_nil ?gFnormal_trans.
rewrite (subset_trans _ (FittingS (pcore_sub _ _))) // subsetI pcore_sub.
by rewrite pcore_max ?pcore_pgroup ?gFnormal_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
|
Fitting_pcore
| |
index_maxnormal_sol_prime(H : {group gT}) :
solvable G -> maxnormal H G G -> prime #|G : H|.
Proof.
move=> solG maxH; have nsHG := maxnormal_normal maxH.
rewrite -card_quotient ?normal_norm // simple_sol_prime ?quotient_sol //.
by rewrite quotient_simple.
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
|
index_maxnormal_sol_prime
| |
sol_prime_factor_exists:
solvable G -> G :!=: 1 -> {H : {group gT} | H <| G & prime #|G : H| }.
Proof.
move=> solG /ex_maxnormal_ntrivg[H maxH].
by exists H; [apply: maxnormal_normal | apply: index_maxnormal_sol_prime].
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
|
sol_prime_factor_exists
| |
center_special_abelem: p.-group G -> special G -> p.-abelem 'Z(G).
Proof.
move=> pG [defPhi defG'].
have [-> | ntG] := eqsVneq G 1; first by rewrite center1 abelem1.
have [p_pr _ _] := pgroup_pdiv pG ntG.
have fM: {in 'Z(G) &, {morph natexp^~ p : x y / x * y}}.
by move=> x y /setIP[_ /centP cxG] /setIP[/cxG cxy _]; apply: expgMn.
rewrite abelemE //= center_abelian; apply/exponentP=> /= z Zz.
apply: (@kerP _ _ _ (Morphism fM)) => //; apply: subsetP z Zz.
rewrite -{1}defG' gen_subG; apply/subsetP=> _ /imset2P[x y Gx Gy ->].
have Zxy: [~ x, y] \in 'Z(G) by rewrite -defG' mem_commg.
have Zxp: x ^+ p \in 'Z(G).
rewrite -defPhi (Phi_joing pG) (MhoE 1 pG) joing_idr mem_gen // !inE.
by rewrite expn1 orbC (imset_f (natexp^~ p)).
rewrite mem_morphpre /= ?defG' ?Zxy // inE -commXg; last first.
by red; case/setIP: Zxy => _ /centP->.
by apply/commgP; red; case/setIP: Zxp => _ /centP->.
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
|
center_special_abelem
| |
exponent_special: p.-group G -> special G -> exponent G %| p ^ 2.
Proof.
move=> pG spG; have [defPhi _] := spG.
have /and3P[_ _ expZ] := center_special_abelem pG spG.
apply/exponentP=> x Gx; rewrite expgM (exponentP expZ) // -defPhi.
by rewrite (Phi_joing pG) mem_gen // inE orbC (Mho_p_elt 1) ?(mem_p_elt pG).
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
|
exponent_special
| |
abelian_charsimple_special:
p.-group G -> coprime #|G| #|A| -> [~: G, A] = G ->
\bigcup_(H : {group gT} | (H \char G) && abelian H) H \subset 'C(A) ->
special G /\ 'C_G(A) = 'Z(G).
Proof.
move=> pG coGA defG /bigcupsP cChaA.
have cZA: 'Z(G) \subset 'C_G(A).
by rewrite subsetI center_sub cChaA // center_char center_abelian.
have cChaG (H : {group gT}): H \char G -> abelian H -> H \subset 'Z(G).
move=> chH abH; rewrite subsetI char_sub //= centsC -defG.
rewrite comm_norm_cent_cent ?(char_norm chH) -?commg_subl ?defG //.
by rewrite centsC cChaA ?chH.
have cZ2GG: [~: 'Z_2(G), G, G] = 1.
by apply/commG1P; rewrite (subset_trans (ucn_comm 1 G)) // ucn1 subsetIr.
have{cZ2GG} cG'Z: 'Z_2(G) \subset 'C(G^`(1)).
by rewrite centsC; apply/commG1P; rewrite three_subgroup // (commGC G).
have{cG'Z} sZ2G'_Z: 'Z_2(G) :&: G^`(1) \subset 'Z(G).
apply: cChaG; first by rewrite charI ?ucn_char ?der_char.
by rewrite /abelian subIset // (subset_trans cG'Z) // centS ?subsetIr.
have{sZ2G'_Z} sG'Z: G^`(1) \subset 'Z(G).
rewrite der1_min ?gFnorm //; apply/derG1P.
have /TI_center_nil: nilpotent (G / 'Z(G)) := quotient_nil _ (pgroup_nil pG).
apply; first exact: gFnormal; rewrite /= setIC -ucn1 -ucn_central.
rewrite -quotient_der ?gFnorm // -quotientGI ?ucn_subS ?quotientS1 //=.
by rewrite ucn1.
have sCG': 'C_G(A) \subset G^`(1).
rewrite -quotient_sub1 //; last by rewrite subIset ?gFnorm.
rewrite (subset_trans (quotient_subcent _ G A)) //= -[G in G / _]defG.
have nGA: A \subset 'N(G) by rewri
...
|
Theorem
|
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
|
abelian_charsimple_special
| |
extraspecial_prime: prime p.
Proof.
by case: esS => _ /prime_gt1; rewrite cardG_gt1; case/(pgroup_pdiv pZ).
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
|
extraspecial_prime
| |
card_center_extraspecial: #|'Z(S)| = p.
Proof. by apply/eqP; apply: (pgroupP pZ); case: esS. 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
|
card_center_extraspecial
| |
min_card_extraspecial: #|S| >= p ^ 3.
Proof.
have p_gt1 := prime_gt1 extraspecial_prime.
rewrite leqNgt (card_pgroup pS) ltn_exp2l // ltnS.
case: esS => [[_ defS']]; apply: contraL => /(p2group_abelian pS)/derG1P S'1.
by rewrite -defS' S'1 cards1.
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
|
min_card_extraspecial
| |
card_p3group_extraspecialE :
prime p -> #|E| = (p ^ 3)%N -> #|'Z(E)| = p -> extraspecial E.
Proof.
move=> p_pr oEp3 oZp; have p_gt0 := prime_gt0 p_pr.
have pE: p.-group E by rewrite /pgroup oEp3 pnatX pnat_id.
have pEq: p.-group (E / 'Z(E))%g by rewrite quotient_pgroup.
have /andP[sZE nZE] := center_normal E.
have oEq: #|E / 'Z(E)|%g = (p ^ 2)%N.
by rewrite card_quotient -?divgS // oEp3 oZp expnS mulKn.
have cEEq: abelian (E / 'Z(E))%g by apply: card_p2group_abelian oEq.
have not_cEE: ~~ abelian E.
have: #|'Z(E)| < #|E| by rewrite oEp3 oZp (ltn_exp2l 1) ?prime_gt1.
by apply: contraL => cEE; rewrite -leqNgt subset_leq_card // subsetI subxx.
have defE': E^`(1) = 'Z(E).
apply/eqP; rewrite eqEsubset der1_min //=; apply: contraR not_cEE => not_sE'Z.
apply/commG1P/(TI_center_nil (pgroup_nil pE) (der_normal 1 _)).
by rewrite setIC prime_TIg ?oZp.
split; [split=> // | by rewrite oZp]; apply/eqP.
rewrite eqEsubset andbC -{1}defE' {1}(Phi_joing pE) joing_subl.
rewrite -quotient_sub1 ?gFsub_trans ?subG1 //=.
rewrite (quotient_Phi pE) //= (trivg_Phi pEq).
apply/abelemP=> //; split=> // Zx EqZx; apply/eqP; rewrite -order_dvdn /order.
rewrite (card_pgroup (mem_p_elt pEq EqZx)) (@dvdn_exp2l _ _ 1) //.
rewrite leqNgt -pfactor_dvdn // -oEq; apply: contra not_cEE => sEqZx.
rewrite cyclic_center_factor_abelian //; apply/cyclicP.
exists Zx; apply/eqP; rewrite eq_sym eqEcard cycle_subG EqZx -orderE.
exact: dvdn_leq sEqZx.
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
|
card_p3group_extraspecial
| |
p3group_extraspecialG :
p.-group G -> ~~ abelian G -> logn p #|G| <= 3 -> extraspecial G.
Proof.
move=> pG not_cGG; have /andP[sZG nZG] := center_normal G.
have ntG: G :!=: 1 by apply: contraNneq not_cGG => ->; apply: abelian1.
have ntZ: 'Z(G) != 1 by rewrite (center_nil_eq1 (pgroup_nil pG)).
have [p_pr _ [n oG]] := pgroup_pdiv pG ntG; rewrite oG pfactorK //.
have [_ _ [m oZ]] := pgroup_pdiv (pgroupS sZG pG) ntZ.
have lt_m1_n: m.+1 < n.
suffices: 1 < logn p #|(G / 'Z(G))|.
rewrite card_quotient // -divgS // logn_div ?cardSg //.
by rewrite oG oZ !pfactorK // ltn_subRL addn1.
rewrite ltnNge; apply: contra not_cGG => cycGs.
apply: cyclic_center_factor_abelian; rewrite (dvdn_prime_cyclic p_pr) //.
by rewrite (card_pgroup (quotient_pgroup _ pG)) (dvdn_exp2l _ cycGs).
rewrite -{lt_m1_n}(subnKC lt_m1_n) !addSn !ltnS leqn0 in oG *.
case: m => // in oZ oG * => /eqP n2; rewrite {n}n2 in oG.
exact: card_p3group_extraspecial oZ.
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
|
p3group_extraspecial
| |
extraspecial_nonabelianG : extraspecial G -> ~~ abelian G.
Proof.
case=> [[_ defG'] oZ]; rewrite /abelian (sameP commG1P eqP).
by rewrite -derg1 defG' -cardG_gt1 prime_gt1.
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
|
extraspecial_nonabelian
| |
exponent_2extraspecialG : 2.-group G -> extraspecial G -> exponent G = 4.
Proof.
move=> p2G esG; have [spG _] := esG.
case/dvdn_pfactor: (exponent_special p2G spG) => // k.
rewrite leq_eqVlt ltnS => /predU1P[-> // | lek1] expG.
case/negP: (extraspecial_nonabelian esG).
by rewrite (@abelem_abelian _ 2) ?exponent2_abelem // expG pfactor_dvdn.
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
|
exponent_2extraspecial
| |
injm_specialD G (f : {morphism D >-> rT}) :
'injm f -> G \subset D -> special G -> special (f @* G).
Proof.
move=> injf sGD [defPhiG defG'].
by rewrite /special -morphim_der // -injm_Phi // defPhiG defG' injm_center.
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_special
| |
injm_extraspecialD G (f : {morphism D >-> rT}) :
'injm f -> G \subset D -> extraspecial G -> extraspecial (f @* G).
Proof.
move=> injf sGD [spG ZG_pr]; split; first exact: injm_special spG.
by rewrite -injm_center // card_injm // subIset ?sGD.
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_extraspecial
| |
isog_specialG (R : {group rT}) :
G \isog R -> special G -> special R.
Proof. by case/isogP=> f injf <-; apply: injm_special. 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_special
| |
isog_extraspecialG (R : {group rT}) :
G \isog R -> extraspecial G -> extraspecial R.
Proof. by case/isogP=> f injf <-; apply: injm_extraspecial. 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_extraspecial
| |
cprod_extraspecialG H K :
p.-group G -> H \* K = G -> H :&: K = 'Z(H) ->
extraspecial H -> extraspecial K -> extraspecial G.
Proof.
move=> pG defG ziHK [[PhiH defH'] ZH_pr] [[PhiK defK'] ZK_pr].
have [_ defHK cHK]:= cprodP defG.
have sZHK: 'Z(H) \subset 'Z(K).
by rewrite subsetI -{1}ziHK subsetIr subIset // centsC cHK.
have{sZHK} defZH: 'Z(H) = 'Z(K).
by apply/eqP; rewrite eqEcard sZHK leq_eqVlt eq_sym -dvdn_prime2 ?cardSg.
have defZ: 'Z(G) = 'Z(K).
by case/cprodP: (center_cprod defG) => /= _ <- _; rewrite defZH mulGid.
split; first split; rewrite defZ //.
by have /cprodP[_ <- _] := Phi_cprod pG defG; rewrite PhiH PhiK defZH mulGid.
by have /cprodP[_ <- _] := der_cprod 1 defG; rewrite defH' defK' defZH mulGid.
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
|
cprod_extraspecial
| |
cent1_extraspecial_maximalx :
x \in G -> x \notin 'Z(G) -> maximal 'C_G[x] G.
Proof.
move=> Gx notZx; pose f y := [~ x, y]; have [[_ defG'] prZ] := esG.
have{defG'} fZ y: y \in G -> f y \in 'Z(G).
by move=> Gy; rewrite -defG' mem_commg.
have fM: {in G &, {morph f : y z / y * z}}%g.
move=> y z Gy Gz; rewrite {1}/f commgMJ conjgCV -conjgM (conjg_fixP _) //.
rewrite (sameP commgP cent1P); apply: subsetP (fZ y Gy).
by rewrite subIset // orbC -cent_set1 centS // sub1set !(groupM, groupV).
pose fm := Morphism fM.
have fmG: fm @* G = 'Z(G).
have sfmG: fm @* G \subset 'Z(G).
by apply/subsetP=> _ /morphimP[z _ Gz ->]; apply: fZ.
apply/eqP; rewrite eqEsubset sfmG; apply: contraR notZx => /(prime_TIg prZ).
rewrite (setIidPr _) // => fmG1; rewrite inE Gx; apply/centP=> y Gy.
by apply/commgP; rewrite -in_set1 -[[set _]]fmG1; apply: mem_morphim.
have ->: 'C_G[x] = 'ker fm.
apply/setP=> z; rewrite inE (sameP cent1P commgP) !inE.
by rewrite -invg_comm eq_invg_mul mulg1.
rewrite p_index_maximal ?subsetIl // -card_quotient ?ker_norm //.
by rewrite (card_isog (first_isog fm)) /= fmG.
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
|
cent1_extraspecial_maximal
| |
subcent1_extraspecial_maximalU x :
U \subset G -> x \in G :\: 'C(U) -> maximal 'C_U[x] U.
Proof.
move=> sUG /setDP[Gx not_cUx]; apply/maxgroupP; split=> [|H ltHU sCxH].
by rewrite /proper subsetIl subsetI subxx sub_cent1.
case/andP: ltHU => sHU not_sHU; have sHG := subset_trans sHU sUG.
apply/eqP; rewrite eqEsubset sCxH subsetI sHU /= andbT.
apply: contraR not_sHU => not_sHCx.
have maxCx: maximal 'C_G[x] G.
rewrite cent1_extraspecial_maximal //; apply: contra not_cUx.
by rewrite inE Gx; apply: subsetP (centS sUG) _.
have nsCx := p_maximal_normal pG maxCx.
rewrite -(setIidPl sUG) -(mulg_normal_maximal nsCx maxCx sHG) ?subsetI ?sHG //.
by rewrite -group_modr //= setIA (setIidPl sUG) mul_subG.
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
|
subcent1_extraspecial_maximal
| |
card_subcent_extraspecialU :
U \subset G -> #|'C_G(U)| = (#|'Z(G) :&: U| * #|G : U|)%N.
Proof.
move=> sUG; rewrite setIAC (setIidPr sUG).
have [m leUm] := ubnP #|U|; elim: m => // m IHm in U leUm sUG *.
have [cUG | not_cUG]:= orP (orbN (G \subset 'C(U))).
by rewrite !(setIidPl _) ?Lagrange // centsC.
have{not_cUG} [x Gx not_cUx] := subsetPn not_cUG.
pose W := 'C_U[x]; have sCW_G: 'C_G(W) \subset G := subsetIl G _.
have maxW: maximal W U by rewrite subcent1_extraspecial_maximal // inE not_cUx.
have nsWU: W <| U := p_maximal_normal (pgroupS sUG pG) maxW.
have ltWU: W \proper U by apply: maxgroupp maxW.
have [sWU [u Uu notWu]] := properP ltWU; have sWG := subset_trans sWU sUG.
have defU: W * <[u]> = U by rewrite (mulg_normal_maximal nsWU) ?cycle_subG.
have iCW_CU: #|'C_G(W) : 'C_G(U)| = p.
rewrite -defU centM cent_cycle setIA /=; rewrite inE Uu cent1C in notWu.
apply: p_maximal_index (pgroupS sCW_G pG) _.
apply: subcent1_extraspecial_maximal sCW_G _.
rewrite inE andbC (subsetP sUG) //= -sub_cent1.
by apply/subsetPn; exists x; rewrite // inE Gx -sub_cent1 subsetIr.
apply/eqP; rewrite -(eqn_pmul2r p_gt0) -{1}iCW_CU Lagrange ?setIS ?centS //.
rewrite IHm ?(leq_trans (proper_card ltWU)) // -setIA -mulnA.
rewrite -(Lagrange_index sUG sWU) (p_maximal_index (pgroupS sUG pG)) //=.
by rewrite -cent_set1 (setIidPr (centS _)) ?sub1set.
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
|
card_subcent_extraspecial
| |
split1_extraspecialx :
x \in G :\: 'Z(G) ->
{E : {group gT} & {R : {group gT} |
[/\ #|E| = (p ^ 3)%N /\ #|R| = #|G| %/ p ^ 2,
E \* R = G /\ E :&: R = 'Z(E),
'Z(E) = 'Z(G) /\ 'Z(R) = 'Z(G),
extraspecial E /\ x \in E
& if abelian R then R :=: 'Z(G) else extraspecial R]}}.
Proof.
case/setDP=> Gx notZx; rewrite inE Gx /= in notZx.
have [[defPhiG defG'] prZ] := esG.
have maxCx: maximal 'C_G[x] G.
by rewrite subcent1_extraspecial_maximal // inE notZx.
pose y := repr (G :\: 'C[x]).
have [Gy not_cxy]: y \in G /\ y \notin 'C[x].
move/maxgroupp: maxCx => /properP[_ [t Gt not_cyt]].
by apply/setDP; apply: (mem_repr t); rewrite !inE Gt andbT in not_cyt *.
pose E := <[x]> <*> <[y]>; pose R := 'C_G(E).
exists [group of E]; exists [group of R] => /=.
have sEG: E \subset G by rewrite join_subG !cycle_subG Gx.
have [Ex Ey]: x \in E /\ y \in E by rewrite !mem_gen // inE cycle_id ?orbT.
have sZE: 'Z(G) \subset E.
rewrite (('Z(G) =P E^`(1)) _) ?der_sub // eqEsubset -{2}defG' dergS // andbT.
apply: contraR not_cxy => /= not_sZE'.
rewrite (sameP cent1P commgP) -in_set1 -[[set 1]](prime_TIg prZ not_sZE').
by rewrite /= -defG' inE !mem_commg.
have ziER: E :&: R = 'Z(E) by rewrite setIA (setIidPl sEG).
have cER: R \subset 'C(E) by rewrite subsetIr.
have iCxG: #|G : 'C_G[x]| = p by apply: p_maximal_index.
have maxR: maximal R 'C_G[x].
rewrite /R centY !cent_cycle setIA.
rewrite subcent1_extraspecial_maximal ?subsetIl // inE Gy andbT -sub_cent1.
by apply/subsetP
...
|
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
|
split1_extraspecial
| |
pmaxElem_extraspecial: 'E*_p(G) = 'E_p^('r_p(G))(G).
Proof.
have sZmax: {in 'E*_p(G), forall E, 'Z(G) \subset E}.
move=> E maxE; have defE := pmaxElem_LdivP p_pr maxE.
have abelZ: p.-abelem 'Z(G) by rewrite prime_abelem ?oZ.
rewrite -(Ohm1_id abelZ) (OhmE 1 (abelem_pgroup abelZ)) gen_subG -defE.
by rewrite setSI // setIS ?centS // -defE !subIset ?subxx.
suffices card_max: {in 'E*_p(G) &, forall E F, #|E| <= #|F| }.
have EprGmax: 'E_p^('r_p(G))(G) \subset 'E*_p(G) := p_rankElem_max p G.
have [E EprE]:= p_rank_witness p G; have maxE := subsetP EprGmax E EprE.
apply/eqP; rewrite eqEsubset EprGmax andbT; apply/subsetP=> F maxF.
rewrite inE; have [-> _]:= pmaxElemP maxF; have [_ _ <-]:= pnElemP EprE.
by apply/eqP; congr (logn p _); apply/eqP; rewrite eqn_leq !card_max.
move=> E F maxE maxF; set U := E :&: F.
have [sUE sUF]: U \subset E /\ U \subset F by apply/andP; rewrite -subsetI.
have sZU: 'Z(G) \subset U by rewrite subsetI !sZmax.
have [EpE _]:= pmaxElemP maxE; have{EpE} [sEG abelE] := pElemP EpE.
have [EpF _]:= pmaxElemP maxF; have{EpF} [sFG abelF] := pElemP EpF.
have [V] := abelem_split_dprod abelE sUE; case/dprodP=> _ defE cUV tiUV.
have [W] := abelem_split_dprod abelF sUF; case/dprodP=> _ defF _ tiUW.
have [sVE sWF]: V \subset E /\ W \subset F by rewrite -defE -defF !mulG_subr.
have [sVG sWG] := (subset_trans sVE sEG, subset_trans sWF sFG).
rewrite -defE -defF !TI_cardMg // leq_pmul2l ?cardG_gt0 //.
rewrite -(leq_pmul2r (cardG_gt0 'C_G(W))) mul_cardG.
rewrite card_subcent_ext
...
|
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
|
pmaxElem_extraspecial
| |
critical_extraspecialR S :
p.-group R -> S \subset R -> extraspecial S -> [~: S, R] \subset S^`(1) ->
S \* 'C_R(S) = R.
Proof.
move=> pR sSR esS sSR_S'; have [[defPhi defS'] _] := esS.
have [pS [sPS nPS]] := (pgroupS sSR pR, andP (Phi_normal S : 'Phi(S) <| S)).
have{esS} oZS: #|'Z(S)| = p := card_center_extraspecial pS esS.
have nSR: R \subset 'N(S) by rewrite -commg_subl (subset_trans sSR_S') ?der_sub.
have nsCR: 'C_R(S) <| R by rewrite (normalGI nSR) ?cent_normal.
have nCS: S \subset 'N('C_R(S)) by rewrite cents_norm // centsC subsetIr.
rewrite cprodE ?subsetIr //= -{2}(quotientGK nsCR) normC -?quotientK //.
congr (_ @*^-1 _); apply/eqP; rewrite eqEcard quotientS //=.
rewrite -(card_isog (second_isog nCS)) setIAC (setIidPr sSR) /= -/'Z(S) -defPhi.
rewrite -ker_conj_aut (card_isog (first_isog_loc _ nSR)) //=; set A := _ @* R.
have{pS} abelSb := Phi_quotient_abelem pS; have [pSb cSSb _] := and3P abelSb.
have [/= Xb defSb oXb] := grank_witness (S / 'Phi(S)).
pose X := (repr \o val : coset_of _ -> gT) @: Xb.
have sXS: X \subset S; last have nPX := subset_trans sXS nPS.
apply/subsetP=> x; case/imsetP=> xb Xxb ->; have nPx := repr_coset_norm xb.
rewrite -sub1set -(quotientSGK _ sPS) ?sub1set ?quotient_set1 //= sub1set.
by rewrite coset_reprK -defSb mem_gen.
have defS: <<X>> = S.
apply: Phi_nongen; apply/eqP; rewrite eqEsubset join_subG sPS sXS -joing_idr.
rewrite -genM_join sub_gen // -quotientSK ?quotient_gen // -defSb genS //.
apply/subsetP=> xb Xxb; apply/imsetP; rewrite (setIidP
...
|
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
|
critical_extraspecial
| |
extraspecial_structureS : p.-group S -> extraspecial S ->
{Es | all (fun E => (#|E| == p ^ 3)%N && ('Z(E) == 'Z(S))) Es
& \big[cprod/1%g]_(E <- Es) E \* 'Z(S) = S}.
Proof.
have [m] := ubnP #|S|; elim: m S => // m IHm S leSm pS esS.
have [x Z'x]: {x | x \in S :\: 'Z(S)}.
apply/sigW/set0Pn; rewrite -subset0 subDset setU0.
apply: contra (extraspecial_nonabelian esS) => sSZ.
exact: abelianS sSZ (center_abelian S).
have [E [R [[oE oR]]]]:= split1_extraspecial pS esS Z'x.
case=> defS _ [defZE defZR] _; case: ifP => [_ defR | _ esR].
by exists [:: E]; rewrite /= ?oE ?defZE ?eqxx // big_seq1 -defR.
have sRS: R \subset S by case/cprodP: defS => _ <- _; rewrite mulG_subr.
have [|Es esEs defR] := IHm _ _ (pgroupS sRS pS) esR.
rewrite oR (leq_trans (ltn_Pdiv _ _)) ?cardG_gt0 // (ltn_exp2l 0) //.
exact: prime_gt1 (extraspecial_prime pS esS).
exists (E :: Es); first by rewrite /= oE defZE !eqxx -defZR.
by rewrite -defZR big_cons -cprodA defR.
Qed.
|
Theorem
|
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_structure
| |
card_extraspecial: {n | n > 0 & #|S| = (p ^ n.*2.+1)%N}.
Proof.
set T := S; exists (logn p #|T|)./2.
rewrite half_gt0 ltnW // -(leq_exp2l _ _ (prime_gt1 p_pr)) -card_pgroup //.
exact: min_card_extraspecial.
have [Es] := extraspecial_structure pS esS; rewrite -[in RHS]/T.
elim: Es T => [_ _ <-| E s IHs T] /=.
by rewrite big_nil cprod1g oZ (pfactorK 1).
rewrite -andbA big_cons -cprodA => /and3P[/eqP oEp3 /eqP defZE].
move=> /IHs{}IHs /cprodP[[_ U _ defU]]; rewrite defU => defT cEU.
rewrite -(mulnK #|T| (cardG_gt0 (E :&: U))) -defT -mul_cardG /=.
have ->: E :&: U = 'Z(S).
apply/eqP; rewrite eqEsubset subsetI -{1 2}defZE subsetIl setIS //=.
by case/cprodP: defU => [[V _ -> _]] <- _; apply: mulG_subr.
rewrite (IHs U) // oEp3 oZ -expnD addSn expnS mulKn ?prime_gt0 //.
by rewrite pfactorK //= uphalf_double.
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
|
card_extraspecial
| |
Aut_extraspecial_full: Aut_in (Aut S) 'Z(S) \isog Aut 'Z(S).
Proof.
have [p_gt1 p_gt0] := (prime_gt1 p_pr, prime_gt0 p_pr).
have [Es] := extraspecial_structure pS esS.
elim: Es S oZ => [T _ _ <-| E s IHs T oZT] /=.
rewrite big_nil cprod1g (center_idP (center_abelian T)).
by apply/Aut_sub_fullP=> // g injg gZ; exists g.
rewrite -andbA big_cons -cprodA => /and3P[/eqP-oE /eqP-defZE es_s].
case/cprodP=> -[_ U _ defU]; rewrite defU => defT cEU.
have sUT: U \subset T by rewrite -defT mulG_subr.
have sZU: 'Z(T) \subset U.
by case/cprodP: defU => [[V _ -> _] <- _]; apply: mulG_subr.
have defZU: 'Z(E) = 'Z(U).
apply/eqP; rewrite eqEsubset defZE subsetI sZU subIset ?centS ?orbT //=.
by rewrite subsetI subIset ?sUT //= -defT centM setSI.
apply: (Aut_cprod_full _ defZU); rewrite ?cprodE //; last first.
by apply: IHs; rewrite -?defZU ?defZE.
have oZE: #|'Z(E)| = p by rewrite defZE.
have [p2 | odd_p] := even_prime p_pr.
suffices <-: restr_perm 'Z(E) @* Aut E = Aut 'Z(E) by apply: Aut_in_isog.
apply/eqP; rewrite eqEcard restr_perm_Aut ?center_sub //=.
by rewrite card_Aut_cyclic ?prime_cyclic ?oZE // {1}p2 cardG_gt0.
have pE: p.-group E by rewrite /pgroup oE pnatX pnat_id.
have nZE: E \subset 'N('Z(E)) by rewrite normal_norm ?center_normal.
have esE: extraspecial E := card_p3group_extraspecial p_pr oE oZE.
have [[defPhiE defE'] prZ] := esE.
have{defPhiE} sEpZ x: x \in E -> (x ^+ p)%g \in 'Z(E).
move=> Ex; rewrite -defPhiE (Phi_joing pE) mem_gen // inE orbC.
by rewrite (Mho_p_elt 1) // (mem_
...
|
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
|
Aut_extraspecial_full
| |
center_aut_extraspecialk : coprime k p ->
exists2 f, f \in Aut S & forall z, z \in 'Z(S) -> f z = (z ^+ k)%g.
Proof.
have /cyclicP[z defZ]: cyclic 'Z(S) by rewrite prime_cyclic ?oZ.
have oz: #[z] = p by rewrite orderE -defZ.
rewrite coprime_sym -unitZpE ?prime_gt1 // -oz => u_k.
pose g := Zp_unitm (FinRing.unit 'Z_#[z] u_k).
have AutZg: g \in Aut 'Z(S) by rewrite defZ -im_Zp_unitm mem_morphim ?inE.
have ZSfull := Aut_sub_fullP (center_sub S) Aut_extraspecial_full.
have [f [injf fS fZ]] := ZSfull _ (injm_autm AutZg) (im_autm AutZg).
exists (aut injf fS) => [|u Zu]; first exact: Aut_aut.
have [Su _] := setIP Zu; have z_u: u \in <[z]> by rewrite -defZ.
by rewrite autE // fZ //= autmE permE /= z_u /cyclem expg_znat.
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
|
center_aut_extraspecial
| |
SCN_PA : reflect (A <| G /\ 'C_G(A) = A) (A \in 'SCN(G)).
Proof. by apply: (iffP setIdP) => [] [->]; move/eqP. 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
|
SCN_P
| |
SCN_abelianA : A \in 'SCN(G) -> abelian A.
Proof. by case/SCN_P=> _ defA; rewrite /abelian -{1}defA subsetIr. 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
|
SCN_abelian
| |
exponent_Ohm1_class2H :
odd p -> p.-group H -> nil_class H <= 2 -> exponent 'Ohm_1(H) %| p.
Proof.
move=> odd_p pH; rewrite nil_class2 => sH'Z; apply/exponentP=> x /=.
rewrite (OhmE 1 pH) expn1 gen_set_id => {x} [/LdivP[] //|].
apply/group_setP; split=> [|x y]; first by rewrite !inE group1 expg1n //=.
case/LdivP=> Hx xp1 /LdivP[Hy yp1]; rewrite !inE groupM //=.
have [_ czH]: [~ y, x] \in H /\ centralises [~ y, x] H.
by apply/centerP; rewrite (subsetP sH'Z) ?mem_commg.
rewrite expMg_Rmul ?xp1 ?yp1 /commute ?czH //= !mul1g.
by rewrite bin2odd // -commXXg ?yp1 /commute ?czH // comm1g.
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
|
exponent_Ohm1_class2
| |
SCN_maxA : A \in 'SCN(G) -> [max A | A <| G & abelian A].
Proof.
case/SCN_P => nAG scA; apply/maxgroupP; split=> [|H].
by rewrite nAG /abelian -{1}scA subsetIr.
do 2![case/andP]=> sHG _ abelH sAH; apply/eqP.
by rewrite eqEsubset sAH -scA subsetI sHG centsC (subset_trans sAH).
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
|
SCN_max
| |
max_SCNA :
p.-group G -> [max A | A <| G & abelian A] -> A \in 'SCN(G).
Proof.
move/pgroup_nil=> nilG; rewrite /abelian.
case/maxgroupP=> /andP[nsAG abelA] maxA; have [sAG nAG] := andP nsAG.
rewrite inE nsAG eqEsubset /= andbC subsetI abelA normal_sub //=.
rewrite -quotient_sub1; last by rewrite subIset 1?normal_norm.
apply/trivgP; apply: (TI_center_nil (quotient_nil A nilG)).
by rewrite quotient_normal // /normal subsetIl normsI ?normG ?norms_cent.
apply/trivgP/subsetP=> _ /setIP[/morphimP[x Nx /setIP[_ Cx]] ->].
rewrite -cycle_subG in Cx => /setIP[GAx CAx].
have{CAx GAx}: <[coset A x]> <| G / A.
by rewrite /normal cycle_subG GAx cents_norm // centsC cycle_subG.
case/(inv_quotientN nsAG)=> B /= defB sAB nBG.
rewrite -cycle_subG defB (maxA B) ?trivg_quotient // nBG.
have{} defB : B :=: A * <[x]>.
rewrite -quotientK ?cycle_subG ?quotient_cycle // defB quotientGK //.
exact: normalS (normal_sub nBG) nsAG.
apply/setIidPl; rewrite ?defB -[_ :&: _]center_prod //=.
rewrite /center !(setIidPl _) //; apply: cycle_abelian.
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
|
max_SCN
| |
der1_stab_Ohm1_SCN_series: ('C(Z) :&: 'C_G(A / Z | 'Q))^`(1) \subset A.
Proof.
case/SCN_P: SCN_A => /andP[sAG nAG] {4} <-.
rewrite subsetI {1}setICA comm_subG ?subsetIl //= gen_subG.
apply/subsetP=> w /imset2P[u v].
rewrite /= -groupV -(groupV _ v) /= astabQR //= -/Z !inE (groupV 'C(Z)).
case/and4P=> cZu _ _ sRuZ /and4P[cZv' _ _ sRvZ] ->{w}.
apply/centP=> a Aa; rewrite /commute -!mulgA (commgCV v) (mulgA u).
rewrite (centP cZu); last by rewrite (subsetP sRvZ) ?mem_commg ?set11 ?groupV.
rewrite 2!(mulgA v^-1) mulKVg 4!mulgA invgK (commgC u^-1) mulgA.
rewrite -(mulgA _ _ v^-1) -(centP cZv') ?(subsetP sRuZ) ?mem_commg ?set11//.
by rewrite -!mulgA invgK mulKVg !mulKg.
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
|
der1_stab_Ohm1_SCN_series
| |
Ohm1_stab_Ohm1_SCN_series:
odd p -> p.-group G -> 'Ohm_1('C_G(Z)) \subset 'C_G(A / Z | 'Q).
Proof.
have [-> | ntG] := eqsVneq G 1; first by rewrite !(setIidPl (sub1G _)) Ohm1.
move=> p_odd pG; have{ntG} [p_pr _ _] := pgroup_pdiv pG ntG.
case/SCN_P: SCN_A => /andP[sAG nAG] _; have pA := pgroupS sAG pG.
have pCGZ : p.-group 'C_G(Z) by rewrite (pgroupS _ pG) // subsetIl.
rewrite {pCGZ}(OhmE 1 pCGZ) gen_subG; apply/subsetP=> x; rewrite /= 3!inE -andbA.
rewrite -!cycle_subG => /and3P[sXG cZX xp1] /=; have cXX := cycle_abelian x.
have nZX := cents_norm cZX; have{nAG} nAX := subset_trans sXG nAG.
pose XA := <[x]> <*> A; pose C := 'C(<[x]> / Z | 'Q); pose CA := A :&: C.
pose Y := <[x]> <*> CA; pose W := 'Ohm_1(Y).
have sXC: <[x]> \subset C by rewrite sub_astabQ nZX (quotient_cents _ cXX).
have defY : Y = <[x]> * CA by rewrite -norm_joinEl // normsI ?nAX ?normsG.
have{nAX} defXA: XA = <[x]> * A := norm_joinEl nAX.
suffices{sXC}: XA \subset Y.
rewrite subsetI sXG /= sub_astabQ nZX centsC defY group_modl //= -/Z -/C.
by rewrite subsetI sub_astabQ defXA quotientMl //= !mulG_subG; case/and4P.
have sZCA: Z \subset CA by rewrite subsetI sZA [C]astabQ sub_cosetpre.
have cZCA: CA \subset 'C(Z) by rewrite subIset 1?(sub_abelian_cent2 cAA).
have sZY: Z \subset Y by rewrite (subset_trans sZCA) ?joing_subr.
have{cZCA cZX} cZY: Y \subset 'C(Z) by rewrite join_subG cZX.
have{cXX nZX} sY'Z : Y^`(1) \subset Z.
rewrite der1_min ?cents_norm //= -/Y defY quotientMl // abelianM /= -/Z -/CA.
rewrite !quotient_abelian //
...
|
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
|
Ohm1_stab_Ohm1_SCN_series
| |
Ohm1_cent_max_normal_abelemZ :
odd p -> p.-group G -> [max Z | Z <| G & p.-abelem Z] -> 'Ohm_1('C_G(Z)) = Z.
Proof.
move=> p_odd pG; set X := 'Ohm_1('C_G(Z)).
case/maxgroupP=> /andP[nsZG abelZ] maxZ.
have [sZG nZG] := andP nsZG; have [_ cZZ expZp] := and3P abelZ.
have{nZG} nsXG: X <| G by rewrite gFnormal_trans ?norm_normalI ?norms_cent.
have cZX : X \subset 'C(Z) by apply/gFsub_trans/subsetIr.
have{sZG expZp} sZX: Z \subset X.
rewrite [X](OhmE 1 (pgroupS _ pG)) ?subsetIl ?sub_gen //.
apply/subsetP=> x Zx; rewrite !inE ?(subsetP sZG) ?(subsetP cZZ) //=.
by rewrite (exponentP expZp).
suffices{sZX} expXp: (exponent X %| p).
apply/eqP; rewrite eqEsubset sZX andbT -quotient_sub1 ?cents_norm //= -/X.
have pGq: p.-group (G / Z) by rewrite quotient_pgroup.
rewrite (TI_center_nil (pgroup_nil pGq)) ?quotient_normal //= -/X setIC.
apply/eqP/trivgPn=> [[Zd]]; rewrite inE -!cycle_subG -cycle_eq1 -subG1 /= -/X.
case/andP=> /sub_center_normal nsZdG.
have{nsZdG} [D defD sZD nsDG] := inv_quotientN nsZG nsZdG; rewrite defD.
have sDG := normal_sub nsDG; have nsZD := normalS sZD sDG nsZG.
rewrite quotientSGK ?quotient_sub1 ?normal_norm //= -/X => sDX /negP[].
rewrite (maxZ D) // nsDG andbA (pgroupS sDG) ?(dvdn_trans (exponentS sDX)) //.
have sZZD: Z \subset 'Z(D) by rewrite subsetI sZD centsC (subset_trans sDX).
by rewrite (cyclic_factor_abelian sZZD) //= -defD cycle_cyclic.
pose normal_abelian := [pred A : {group gT} | A <| G & abelian A].
have{nsZG cZZ} normal_abelian_Z : normal_abelian Z
...
|
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
|
Ohm1_cent_max_normal_abelem
| |
critical_class2H : critical H G -> nil_class H <= 2.
Proof.
case=> [chH _ sRZ _].
by rewrite nil_class2 (subset_trans _ sRZ) ?commSg // char_sub.
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
|
critical_class2
| |
Thompson_critical: p.-group G -> {K : {group gT} | critical K G}.
Proof.
move=> pG; pose qcr A := (A \char G) && ('Phi(A) :|: [~: G, A] \subset 'Z(A)).
have [|K]:= @maxgroup_exists _ qcr 1 _.
by rewrite /qcr char1 center1 commG1 subUset Phi_sub subxx.
case/maxgroupP; rewrite {}/qcr subUset => /and3P[chK sPhiZ sRZ] maxK _.
have sKG := char_sub chK; have nKG := char_normal chK.
exists K; split=> //; apply/eqP; rewrite eqEsubset andbC setSI //=.
have chZ: 'Z(K) \char G by [apply: subcent_char]; have nZG := char_norm chZ.
have chC: 'C_G(K) \char G by apply: subcent_char chK.
rewrite -quotient_sub1; last by rewrite subIset // char_norm.
apply/trivgP; apply: (TI_center_nil (quotient_nil _ (pgroup_nil pG))).
by rewrite quotient_normal ?norm_normalI ?norms_cent ?normal_norm.
apply: TI_Ohm1; apply/trivgP; rewrite -trivg_quotient -sub_cosetpre_quo //.
rewrite morphpreI quotientGK /=; last first.
by apply: normalS (char_normal chZ); rewrite ?subsetIl ?setSI.
set X := _ :&: _; pose gX := [group of X].
have sXG: X \subset G by rewrite subIset ?subsetIl.
have cXK: K \subset 'C(gX) by rewrite centsC 2?subIset // subxx orbT.
rewrite subsetI centsC cXK andbT -(mul1g K) -mulSG mul1g -(cent_joinEr cXK).
rewrite [_ <*> K]maxK ?joing_subr //= andbC (cent_joinEr cXK).
rewrite -center_prod // (subset_trans _ (mulG_subr _ _)).
rewrite charM 1?charI ?(char_from_quotient (normal_cosetpre _)) //.
by rewrite cosetpreK !gFchar_trans.
rewrite (@Phi_mulg p) ?(pgroupS _ pG) // subUset commGC commMG; last first.
...
|
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
|
Thompson_critical
| |
critical_p_stab_AutH :
critical H G -> p.-group G -> p.-group 'C(H | [Aut G]).
Proof.
move=> [chH sPhiZ sRZ eqCZ] pG; have sHG := char_sub chH.
pose G' := (sdpair1 [Aut G] @* G)%G; pose H' := (sdpair1 [Aut G] @* H)%G.
apply/pgroupP=> q pr_q; case/Cauchy=> //= f cHF; move: (cHF); rewrite astab_ract.
case/setIP=> Af cHFP ofq; rewrite -cycle_subG in cHF; apply: (pgroupP pG) => //.
pose F' := (sdpair2 [Aut G] @* <[f]>)%G.
have trHF: [~: H', F'] = 1.
apply/trivgP; rewrite gen_subG; apply/subsetP=> u; case/imset2P=> x' a'.
case/morphimP=> x Gx Hx ->; case/morphimP=> a Aa Fa -> -> {u x' a'}.
by rewrite inE commgEl -sdpair_act ?(astab_act (subsetP cHF _ Fa) Hx) ?mulVg.
have sGH_H: [~: G', H'] \subset H'.
by rewrite -morphimR ?(char_sub chH) // morphimS // commg_subr char_norm.
have{trHF sGH_H} trFGH: [~: F', G', H'] = 1.
apply: three_subgroup; last by rewrite trHF comm1G.
by apply/trivgP; rewrite -trHF commSg.
apply/negP=> qG; case: (qG); rewrite -ofq.
suffices ->: f = 1 by rewrite order1 dvd1n.
apply/permP=> x; rewrite perm1; case Gx: (x \in G); last first.
by apply: out_perm (negbT Gx); case/setIdP: Af.
have Gfx: f x \in G by rewrite -(im_autm Af) -{1}(autmE Af) mem_morphim.
pose y := x^-1 * f x; have Gy: y \in G by rewrite groupMl ?groupV.
have [inj1 inj2] := (injm_sdpair1 [Aut G], injm_sdpair2 [Aut G]).
have Hy: y \in H.
rewrite (subsetP (center_sub H)) // -eqCZ -cycle_subG.
rewrite -(injmSK inj1) ?cycle_subG // injm_subcent // subsetI.
rewrite morphimS ?morphim_cycle ?cycle_s
...
|
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
|
critical_p_stab_Aut
| |
lower_central_at:= iter n.-1 (fun B => [~: B, A]) A.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lower_central_at
| |
upper_central_at:= iter n (fun B => coset B @*^-1 'Z(A / B)) 1.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
upper_central_at
| |
nilpotent:=
[forall (G : {group gT} | G \subset A :&: [~: G, A]), G :==: 1].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
nilpotent
| |
nil_class:= index 1 (mkseq (fun n => 'L_n.+1(A)) #|A|).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
nil_class
| |
solvable:=
[forall (G : {group gT} | G \subset A :&: [~: G, G]), G :==: 1].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
solvable
| |
nilpotent1: nilpotent [1 gT].
Proof. by apply/forall_inP=> H; rewrite commG1 setIid -subG1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
nilpotent1
| |
nilpotentSA B : B \subset A -> nilpotent A -> nilpotent B.
Proof.
move=> sBA nilA; apply/forall_inP=> H sHR.
have:= forallP nilA H; rewrite (subset_trans sHR) //.
by apply: subset_trans (setIS _ _) (setSI _ _); rewrite ?commgS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
nilpotentS
| |
nil_comm_properlG H A :
nilpotent G -> H \subset G -> H :!=: 1 -> A \subset 'N_G(H) ->
[~: H, A] \proper H.
Proof.
move=> nilG sHG ntH; rewrite subsetI properE; case/andP=> sAG nHA.
rewrite (subset_trans (commgS H (subset_gen A))) ?commg_subl ?gen_subG //.
apply: contra ntH => sHR; have:= forallP nilG H; rewrite subsetI sHG.
by rewrite (subset_trans sHR) ?commgS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
nil_comm_properl
| |
nil_comm_properrG A H :
nilpotent G -> H \subset G -> H :!=: 1 -> A \subset 'N_G(H) ->
[~: A, H] \proper H.
Proof. by rewrite commGC; apply: nil_comm_properl. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
nil_comm_properr
| |
centrals_nil(s : seq {group gT}) G :
G.-central.-series 1%G s -> last 1%G s = G -> nilpotent G.
Proof.
move=> cGs defG; apply/forall_inP=> H /subsetIP[sHG sHR].
move: sHG; rewrite -{}defG -subG1 -[1]/(gval 1%G).
elim: s 1%G cGs => //= L s IHs K /andP[/and3P[sRK sKL sLG] /IHs sHL] sHs.
exact: subset_trans sHR (subset_trans (commSg _ (sHL sHs)) sRK).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
centrals_nil
| |
lcn0A : 'L_0(A) = A. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn0
| |
lcn1A : 'L_1(A) = A. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn1
| |
lcnSnn A : 'L_n.+2(A) = [~: 'L_n.+1(A), A]. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcnSn
| |
lcnSnSn G : [~: 'L_n(G), G] \subset 'L_n.+1(G).
Proof. by case: n => //; apply: der1_subG. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcnSnS
| |
lcnEn A : 'L_n.+1(A) = iter n (fun B => [~: B, A]) A.
Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcnE
| |
lcn2A : 'L_2(A) = A^`(1). Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn2
| |
lcn_group_setn G : group_set 'L_n(G).
Proof. by case: n => [|[|n]]; apply: groupP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_group_set
| |
lower_central_at_groupn G := Group (lcn_group_set n G).
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lower_central_at_group
| |
lcn_charn G : 'L_n(G) \char G.
Proof. by case: n; last elim=> [|n IHn]; rewrite ?char_refl ?lcnSn ?charR. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_char
| |
lcn_normaln G : 'L_n(G) <| G.
Proof. exact/char_normal/lcn_char. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_normal
| |
lcn_subn G : 'L_n(G) \subset G.
Proof. exact/char_sub/lcn_char. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_sub
| |
lcn_normn G : G \subset 'N('L_n(G)).
Proof. exact/char_norm/lcn_char. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_norm
| |
lcn_subSn G : 'L_n.+1(G) \subset 'L_n(G).
Proof.
case: n => // n; rewrite lcnSn commGC commg_subr.
by case/andP: (lcn_normal n.+1 G).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_subS
| |
lcn_normalSn G : 'L_n.+1(G) <| 'L_n(G).
Proof. by apply: normalS (lcn_normal _ _); rewrite (lcn_subS, lcn_sub). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_normalS
| |
lcn_centraln G : 'L_n(G) / 'L_n.+1(G) \subset 'Z(G / 'L_n.+1(G)).
Proof.
case: n => [|n]; first by rewrite trivg_quotient sub1G.
by rewrite subsetI quotientS ?lcn_sub ?quotient_cents2r.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_central
| |
lcn_sub_leqm n G : n <= m -> 'L_m(G) \subset 'L_n(G).
Proof.
by move/subnK <-; elim: {m}(m - n) => // m; apply: subset_trans (lcn_subS _ _).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_sub_leq
| |
lcnSn A B : A \subset B -> 'L_n(A) \subset 'L_n(B).
Proof.
by case: n => // n sAB; elim: n => // n IHn; rewrite !lcnSn genS ?imset2S.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcnS
| |
lcn_cprodn A B G : A \* B = G -> 'L_n(A) \* 'L_n(B) = 'L_n(G).
Proof.
case: n => // n /cprodP[[H K -> ->{A B}] defG cHK].
have sL := subset_trans (lcn_sub _ _); rewrite cprodE ?(centSS _ _ cHK) ?sL //.
symmetry; elim: n => // n; rewrite lcnSn => ->; rewrite commMG /=; last first.
by apply: subset_trans (commg_normr _ _); rewrite sL // -defG mulG_subr.
rewrite -!(commGC G) -defG -{1}(centC cHK).
rewrite !commMG ?normsR ?lcn_norm ?cents_norm // 1?centsC //.
by rewrite -!(commGC 'L__(_)) -!lcnSn !(commG1P _) ?mul1g ?sL // centsC.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_cprod
| |
lcn_dprodn A B G : A \x B = G -> 'L_n(A) \x 'L_n(B) = 'L_n(G).
Proof.
move=> defG; have [[K H defA defB] _ _ tiAB] := dprodP defG.
rewrite !dprodEcp // in defG *; first exact: lcn_cprod.
by rewrite defA defB; apply/trivgP; rewrite -tiAB defA defB setISS ?lcn_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
lcn_dprod
| |
der_cprodn A B G : A \* B = G -> A^`(n) \* B^`(n) = G^`(n).
Proof. by move=> defG; elim: n => {defG}// n; apply: (lcn_cprod 2). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
der_cprod
| |
der_dprodn A B G : A \x B = G -> A^`(n) \x B^`(n) = G^`(n).
Proof. by move=> defG; elim: n => {defG}// n; apply: (lcn_dprod 2). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From mathcomp Require finfun"
] |
solvable/nilpotent.v
|
der_dprod
|
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