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 |
|---|---|---|---|---|---|---|
gFnormalgT (G : {group gT}) : F gT G <| G.
Proof. exact/char_normal/gFchar. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFnormal
| |
gFchar_transgT (G H : {group gT}) : H \char G -> F gT H \char G.
Proof. exact/char_trans/gFchar. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFchar_trans
| |
gFnormal_transgT (G H : {group gT}) : H <| G -> F gT H <| G.
Proof. exact/char_normal_trans/gFchar. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFnormal_trans
| |
gFnorm_transgT (A : {pred gT}) (G : {group gT}) :
A \subset 'N(G) -> A \subset 'N(F gT G).
Proof. by move/subset_trans/(_ (gFnorms G)). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFnorm_trans
| |
injmF_subgT rT (G D : {group gT}) (f : {morphism D >-> rT}) :
'injm f -> G \subset D -> f @* (F gT G) \subset F rT (f @* G).
Proof.
move=> injf sGD; have:= gFiso_cont (injm_restrm sGD injf).
by rewrite im_restrm morphim_restrm (setIidPr _) ?gFsub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
injmF_sub
| |
injmFgT rT (G D : {group gT}) (f : {morphism D >-> rT}) :
'injm f -> G \subset D -> f @* (F gT G) = F rT (f @* G).
Proof.
move=> injf sGD; have [sfGD injf'] := (morphimS f sGD, injm_invm injf).
apply/esym/eqP; rewrite eqEsubset -(injmSK injf') ?gFsub_trans //.
by rewrite !(subset_trans (injmF_sub _ _)) ?morphim_invm // gFsub_trans.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
injmF
| |
gFisomgT rT (G D : {group gT}) R (f : {morphism D >-> rT}) :
G \subset D -> isom G (gval R) f -> isom (F gT G) (F rT R) f.
Proof.
case/(restrmP f)=> g [gf _ _ _]; rewrite -{f}gf => /isomP[injg <-].
by rewrite sub_isom ?gFsub ?injmF.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFisom
| |
gFisoggT rT (G : {group gT}) (R : {group rT}) :
G \isog R -> F gT G \isog F rT R.
Proof. by case/isogP=> f injf <-; rewrite -injmF // sub_isog ?gFsub. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFisog
| |
gFcont: GFunctor.continuous F.
Proof. by case F. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFcont
| |
morphimFgT rT (G D : {group gT}) (f : {morphism D >-> rT}) :
G \subset D -> f @* (F gT G) \subset F rT (f @* G).
Proof.
move=> sGD; rewrite -(setIidPr (gFsub F G)).
by rewrite -{3}(setIid G) -!(morphim_restrm sGD) gFcont.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
morphimF
| |
gFhereditary: GFunctor.hereditary F.
Proof. by case F. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFhereditary
| |
gFunctorIgT (G H : {group gT}) :
F gT G :&: H = F gT G :&: F gT (G :&: H).
Proof.
rewrite -{1}(setIidPr (gFsub F G)) setIAC setIC.
rewrite -(setIidPr (gFhereditary (subsetIl G H))).
by rewrite setIC -setIA (setIidPr (gFsub F (G :&: H))).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFunctorI
| |
pmorphimF: GFunctor.pcontinuous F.
Proof.
move=> gT rT G D f; rewrite -morphimIdom -(setIidPl (gFsub F G)) setICA.
apply: (subset_trans (morphimS f (gFhereditary (subsetIr D G)))).
by rewrite (subset_trans (morphimF F _ _ )) ?morphimIdom ?subsetIl.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
pmorphimF
| |
gFidgT (G : {group gT}) : F gT (F gT G) = F gT G.
Proof.
apply/eqP; rewrite eqEsubset gFsub.
by move/gFhereditary: (gFsub F G); rewrite setIid /=.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFid
| |
gFmod_closed: GFunctor.closed (F1 %% F2).
Proof. by move=> gT G; rewrite sub_cosetpre_quo ?gFsub ?gFnormal. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFmod_closed
| |
gFmod_cont: GFunctor.continuous (F1 %% F2).
Proof.
move=> gT rT G f; have nF2 := gFnorm F2.
have sDF: G \subset 'dom (coset (F2 _ G)) by rewrite nF2.
have sDFf: G \subset 'dom (coset (F2 _ (f @* G)) \o f).
by rewrite -sub_morphim_pre ?subsetIl // nF2.
pose K := 'ker (restrm sDFf (coset (F2 _ (f @* G)) \o f)).
have sFK: 'ker (restrm sDF (coset (F2 _ G))) \subset K.
rewrite {}/K !ker_restrm ker_comp /= subsetI subsetIl !ker_coset /=.
by rewrite -sub_morphim_pre ?subsetIl // morphimIdom ?morphimF.
have sOF := gFsub F1 (G / F2 _ G); have sGG: G \subset G by [].
rewrite -sub_quotient_pre; last first.
by apply: subset_trans (nF2 _ _); rewrite morphimS ?gFmod_closed.
suffices im_fact H : F2 _ G \subset gval H -> H \subset G ->
factm sFK sGG @* (H / F2 _ G) = f @* H / F2 _ (f @* G).
- rewrite -2?im_fact ?gFmod_closed ?gFsub //.
by rewrite cosetpreK morphimF /= ?morphim_restrm ?setIid.
by rewrite -sub_quotient_pre ?normG //= trivg_quotient sub1G.
move=> sFH sHG; rewrite -(morphimIdom _ (H / _)) /= {2}morphim_restrm /= setIid.
rewrite -morphimIG ?ker_coset // -(morphim_restrm sDF) morphim_factm.
by rewrite morphim_restrm morphim_comp -quotientE morphimIdom.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFmod_cont
| |
gFmod_igFun:= [igFun by gFmod_closed & gFmod_cont].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFmod_igFun
| |
gFmod_gFun:= [gFun by gFmod_cont].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFmod_gFun
| |
gFmod_hereditary: GFunctor.hereditary (F1 %% F2).
Proof.
move=> gT H G sHG; set FGH := _ :&: H; have nF2H := gFnorm F2 H.
rewrite -sub_quotient_pre; last exact: subset_trans (subsetIr _ _) _.
pose rH := restrm nF2H (coset (F2 _ H)); pose rHM := [morphism of rH].
have rnorm_simpl: rHM @* H = H / F2 _ H by rewrite morphim_restrm setIid.
have nF2G := subset_trans sHG (gFnorm F2 G).
pose rG := restrm nF2G (coset (F2 _ G)); pose rGM := [morphism of rG].
have sqKfK: 'ker rGM \subset 'ker rHM.
rewrite !ker_restrm !ker_coset (setIidPr (gFsub F2 _)) setIC /=.
exact: gFhereditary.
have sHH := subxx H; rewrite -rnorm_simpl /= -(morphim_factm sqKfK sHH) /=.
apply: subset_trans (gFcont F1 _); rewrite /= {2}morphim_restrm setIid /=.
apply: subset_trans (morphimS _ (gFhereditary _ (quotientS _ sHG))) => /=.
have ->: FGH / _ = restrm nF2H (coset _) @* FGH.
by rewrite morphim_restrm setICA setIid.
rewrite -(morphim_factm sqKfK sHH) morphimS //= morphim_restrm -quotientE.
by rewrite setICA setIid (subset_trans (quotientI _ _ _)) // cosetpreK.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFmod_hereditary
| |
gFmod_pgFun:= [pgFun by gFmod_hereditary].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFmod_pgFun
| |
gFunctorS(F : GFunctor.mono_map) : GFunctor.monotonic F.
Proof. by case: F. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFunctorS
| |
gFcomp_closed: GFunctor.closed (F1 \o F2).
Proof. by move=> gT G; rewrite !gFsub_trans. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFcomp_closed
| |
gFcomp_cont: GFunctor.continuous (F1 \o F2).
Proof.
move=> gT rT G phi; rewrite (subset_trans (morphimF _ _ (gFsub _ _))) //.
by rewrite (subset_trans (gFunctorS F1 (gFcont F2 phi))).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFcomp_cont
| |
gFcomp_igFun:= [igFun by gFcomp_closed & gFcomp_cont].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFcomp_igFun
| |
gFcomp_gFun:=[gFun by gFcomp_cont].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFcomp_gFun
| |
gFcompS: GFunctor.monotonic (F1 \o F2).
Proof. by move=> gT H G sHG; rewrite !gFunctorS. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFcompS
| |
gFcomp_mgFun:= [mgFun by gFcompS].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFcomp_mgFun
| |
idGfungT := @id {set gT}.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
idGfun
| |
idGfun_closed: GFunctor.closed idGfun. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
idGfun_closed
| |
idGfun_cont: GFunctor.continuous idGfun. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
idGfun_cont
| |
idGfun_monotonic: GFunctor.monotonic idGfun. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
idGfun_monotonic
| |
bgFunc_id:= [igFun by idGfun_closed & idGfun_cont].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
bgFunc_id
| |
gFunc_id:= [gFun by idGfun_cont].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
gFunc_id
| |
mgFunc_id:= [mgFun by idGfun_monotonic].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
mgFunc_id
| |
trivGfungT of {set gT} := [1 gT].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
trivGfun
| |
trivGfun_cont: GFunctor.pcontinuous trivGfun.
Proof. by move=> gT rT D G f; rewrite morphim1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
trivGfun_cont
| |
trivGfun_igFun:= [igFun by sub1G & trivGfun_cont].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
trivGfun_igFun
| |
trivGfun_gFun:= [gFun by trivGfun_cont].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
trivGfun_gFun
| |
trivGfun_pgFun:= [pgFun by trivGfun_cont].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset fingroup morphism automorphism",
"From mathcomp Require Import quotient gproduct"
] |
solvable/gfunctor.v
|
trivGfun_pgFun
| |
subnormalA B :=
(A \subset B) && (iter #|B| (fun N => generated (class_support A N)) B == A).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
subnormal
| |
invariant_factorA B C :=
[&& A \subset 'N(B), A \subset 'N(C) & B <| C].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
invariant_factor
| |
group_rel_of(r : rel {set gT}) := [rel H G : groupT | r H G].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
group_rel_of
| |
stable_factorA V U :=
([~: U, A] \subset V) && (V <| U).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
stable_factor
| |
central_factorA V U :=
[&& [~: U, A] \subset V, V \subset U & U \subset A].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
central_factor
| |
maximalA B := [max A of G | G \proper B].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maximal
| |
maximal_eqA B := (A == B) || maximal A B.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maximal_eq
| |
maxnormalA B U := [max A of G | G \proper B & U \subset 'N(G)].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maxnormal
| |
minnormalA B := [min A of G | G :!=: 1 & B \subset 'N(G)].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
minnormal
| |
simpleA := minnormal A A.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
simple
| |
chief_factorA V U := maxnormal V U A && (U <| A).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
chief_factor
| |
subnormalPH G :
reflect (exists2 s, normal.-series H s & last H s = G) (H <|<| G).
Proof.
apply: (iffP andP) => [[sHG snHG] | [s Hsn <-{G}]].
move: #|G| snHG => m; elim: m => [|m IHm] in G sHG *.
by exists [::]; last by apply/eqP; rewrite eq_sym.
rewrite iterSr => /IHm[|s Hsn defG].
by rewrite sub_gen // class_supportEr (bigD1 1) //= conjsg1 subsetUl.
exists (rcons s G); rewrite ?last_rcons // -cats1 cat_path Hsn defG /=.
rewrite /normal gen_subG class_support_subG //=.
by rewrite norms_gen ?class_support_norm.
set f := fun _ => <<_>>; have idf: iter _ f H == H.
by elim=> //= m IHm; rewrite (eqP IHm) /f class_support_id genGid.
have [m] := ubnP (size s); elim: m s Hsn => // m IHm /lastP[//|s G].
rewrite size_rcons last_rcons rcons_path /= ltnS.
set K := last H s => /andP[Hsn /andP[sKG nKG]] lt_s_m.
have /[1!subEproper]/predU1P[<-|prKG] := sKG; first exact: IHm.
pose L := [group of f G].
have sHK: H \subset K by case/IHm: Hsn.
have sLK: L \subset K by rewrite gen_subG class_support_sub_norm.
rewrite -(subnK (proper_card (sub_proper_trans sLK prKG))) iterD iterSr.
have defH: H = setIgr L H by rewrite -sub_setIgr ?sub_gen ?sub_class_support.
have: normal.-series H (map (setIgr L) s) by rewrite defH path_setIgr.
case/IHm=> [|_]; first by rewrite size_map.
rewrite [in last _]defH last_map (subset_trans sHK) //=.
by rewrite (setIidPr sLK) => /eqP->.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
subnormalP
| |
subnormal_reflG : G <|<| G.
Proof. by apply/subnormalP; exists [::]. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
subnormal_refl
| |
subnormal_transK H G : H <|<| K -> K <|<| G -> H <|<| G.
Proof.
case/subnormalP=> [s1 Hs1 <-] /subnormalP[s2 Hs12 <-].
by apply/subnormalP; exists (s1 ++ s2); rewrite ?last_cat // cat_path Hs1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
subnormal_trans
| |
normal_subnormalH G : H <| G -> H <|<| G.
Proof. by move=> nsHG; apply/subnormalP; exists [:: G]; rewrite //= nsHG. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
normal_subnormal
| |
setI_subnormalG H K : K \subset G -> H <|<| G -> H :&: K <|<| K.
Proof.
move=> sKG /subnormalP[s Hs defG]; apply/subnormalP.
exists (map (setIgr K) s); first exact: path_setIgr.
rewrite (last_map (setIgr K)) defG.
by apply: val_inj; rewrite /= (setIidPr sKG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
setI_subnormal
| |
subnormal_subG H : H <|<| G -> H \subset G.
Proof. by case/andP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
subnormal_sub
| |
invariant_subnormalA G H :
A \subset 'N(G) -> A \subset 'N(H) -> H <|<| G ->
exists2 s, (A.-invariant).-series H s & last H s = G.
Proof.
move=> nGA nHA /andP[]; move: #|G| => m.
elim: m => [|m IHm] in G nGA * => sHG.
by rewrite eq_sym; exists [::]; last apply/eqP.
rewrite iterSr; set K := <<_>>.
have nKA: A \subset 'N(K) by rewrite norms_gen ?norms_class_support.
have sHK: H \subset K by rewrite sub_gen ?sub_class_support.
case/IHm=> // s Hsn defK; exists (rcons s G); last by rewrite last_rcons.
rewrite rcons_path Hsn !andbA defK nGA nKA /= -/K.
by rewrite gen_subG class_support_subG ?norms_gen ?class_support_norm.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
invariant_subnormal
| |
subnormalEsupportG H :
H <|<| G -> H :=: G \/ <<class_support H G>> \proper G.
Proof.
case/andP=> sHG; set K := <<_>> => /eqP <-.
have: K \subset G by rewrite gen_subG class_support_subG.
rewrite subEproper; case/predU1P=> [defK|]; [left | by right].
by elim: #|G| => //= _ ->.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
subnormalEsupport
| |
subnormalErG H : H <|<| G ->
H :=: G \/ (exists K : {group gT}, [/\ H <|<| K, K <| G & K \proper G]).
Proof.
case/subnormalP=> s Hs <-{G}.
elim/last_ind: s Hs => [|s G IHs]; first by left.
rewrite last_rcons -cats1 cat_path /= andbT; set K := last H s.
case/andP=> Hs nsKG; have /[1!subEproper] := normal_sub nsKG.
case/predU1P=> [<- | prKG]; [exact: IHs | right; exists K; split=> //].
by apply/subnormalP; exists s.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
subnormalEr
| |
subnormalElG H : H <|<| G ->
H :=: G \/ (exists K : {group gT}, [/\ H <| K, K <|<| G & H \proper K]).
Proof.
case/subnormalP=> s Hs <-{G}; elim: s H Hs => /= [|K s IHs] H; first by left.
case/andP=> nsHK Ks; have /[1!subEproper] := normal_sub nsHK.
case/predU1P=> [-> | prHK]; [exact: IHs | right; exists K; split=> //].
by apply/subnormalP; exists s.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
subnormalEl
| |
morphim_subnormal(rT : finGroupType) G (f : {morphism G >-> rT}) H K :
H <|<| K -> f @* H <|<| f @* K.
Proof.
case/subnormalP => s Hs <-{K}; apply/subnormalP.
elim: s H Hs => [|K s IHs] H /=; first by exists [::].
case/andP=> nsHK /IHs[fs Hfs <-].
by exists ([group of f @* K] :: fs); rewrite /= ?morphim_normal.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
morphim_subnormal
| |
quotient_subnormalH G K : G <|<| K -> G / H <|<| K / H.
Proof. exact: morphim_subnormal. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
quotient_subnormal
| |
maximal_eqPM G :
reflect (M \subset G /\
forall H, M \subset H -> H \subset G -> H :=: M \/ H :=: G)
(maximal_eq M G).
Proof.
rewrite subEproper /maximal_eq; case: eqP => [->|_]; first left.
by split=> // H sGH sHG; right; apply/eqP; rewrite eqEsubset sHG.
apply: (iffP maxgroupP) => [] [sMG maxM]; split=> // H.
by move/maxM=> maxMH; rewrite subEproper; case/predU1P; auto.
by rewrite properEneq => /andP[/eqP neHG sHG] /maxM[].
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maximal_eqP
| |
maximal_existsH G :
H \subset G ->
H :=: G \/ (exists2 M : {group gT}, maximal M G & H \subset M).
Proof.
rewrite subEproper; case/predU1P=> sHG; first by left.
suff [M *]: {M : {group gT} | maximal M G & H \subset M} by right; exists M.
exact: maxgroup_exists.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maximal_exists
| |
mulg_normal_maximalG M H :
M <| G -> maximal M G -> H \subset G -> ~~ (H \subset M) -> (M * H = G)%g.
Proof.
case/andP=> sMG nMG /maxgroupP[_ maxM] sHG not_sHM.
apply/eqP; rewrite eqEproper mul_subG // -norm_joinEr ?(subset_trans sHG) //.
by apply: contra not_sHM => /maxM <-; rewrite ?joing_subl ?joing_subr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
mulg_normal_maximal
| |
minnormal_existsG H : H :!=: 1 -> G \subset 'N(H) ->
{M : {group gT} | minnormal M G & M \subset H}.
Proof. by move=> ntH nHG; apply: mingroup_exists (H) _; rewrite ntH. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
minnormal_exists
| |
morphpre_maximal: maximal (f @*^-1 M) (f @*^-1 G) = maximal M G.
Proof.
apply/maxgroupP/maxgroupP; rewrite morphpre_proper //= => [] [ltMG maxM].
split=> // H ltHG sMH; have dH := subset_trans (proper_sub ltHG) dG.
rewrite -(morphpreK dH) [f @*^-1 H]maxM ?morphpreK ?morphpreSK //.
by rewrite morphpre_proper.
split=> // H ltHG sMH.
have dH: H \subset D := subset_trans (proper_sub ltHG) (subsetIl D _).
have defH: f @*^-1 (f @* H) = H.
by apply: morphimGK dH; apply: subset_trans sMH; apply: ker_sub_pre.
rewrite -defH morphpre_proper ?morphimS // in ltHG.
by rewrite -defH [f @* H]maxM // -(morphpreK dM) morphimS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
morphpre_maximal
| |
morphpre_maximal_eq: maximal_eq (f @*^-1 M) (f @*^-1 G) = maximal_eq M G.
Proof. by rewrite /maximal_eq morphpre_maximal !eqEsubset !morphpreSK. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
morphpre_maximal_eq
| |
injm_maximal: maximal (f @* M) (f @* G) = maximal M G.
Proof.
rewrite -(morphpre_invm injf) -(morphpre_invm injf G).
by rewrite morphpre_maximal ?morphim_invm.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
injm_maximal
| |
injm_maximal_eq: maximal_eq (f @* M) (f @* G) = maximal_eq M G.
Proof. by rewrite /maximal_eq injm_maximal // injm_eq. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
injm_maximal_eq
| |
injm_maxnormal: maxnormal (f @* M) (f @* G) (f @* L) = maxnormal M G L.
Proof.
pose injfm := (injm_proper injf, injm_norms, injmSK injf, subsetIl).
apply/maxgroupP/maxgroupP; rewrite !injfm // => [[nML maxM]].
split=> // H nHL sMH; have [/proper_sub sHG _] := andP nHL.
have dH := subset_trans sHG dG; apply: (injm_morphim_inj injf) => //.
by apply: maxM; rewrite !injfm.
split=> // fH nHL sMH; have [/proper_sub sfHG _] := andP nHL.
have{sfHG} dfH: fH \subset f @* D := subset_trans sfHG (morphim_sub f G).
by rewrite -(morphpreK dfH) !injfm // in nHL sMH *; rewrite (maxM _ nHL).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
injm_maxnormal
| |
injm_minnormal: minnormal (f @* M) (f @* G) = minnormal M G.
Proof.
pose injfm := (morphim_injm_eq1 injf, injm_norms, injmSK injf, subsetIl).
apply/mingroupP/mingroupP; rewrite !injfm // => [[nML minM]].
split=> // H nHG sHM; have dH := subset_trans sHM dM.
by apply: (injm_morphim_inj injf) => //; apply: minM; rewrite !injfm.
split=> // fH nHG sHM; have dfH := subset_trans sHM (morphim_sub f M).
by rewrite -(morphpreK dfH) !injfm // in nHG sHM *; rewrite (minM _ nHG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
injm_minnormal
| |
cosetpre_maximal(Q R : {group coset_of K}) :
maximal (coset K @*^-1 Q) (coset K @*^-1 R) = maximal Q R.
Proof. by rewrite morphpre_maximal ?sub_im_coset. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
cosetpre_maximal
| |
cosetpre_maximal_eq(Q R : {group coset_of K}) :
maximal_eq (coset K @*^-1 Q) (coset K @*^-1 R) = maximal_eq Q R.
Proof. by rewrite /maximal_eq !eqEsubset !cosetpreSK cosetpre_maximal. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
cosetpre_maximal_eq
| |
quotient_maximal:
K <| G -> K <| H -> maximal (G / K) (H / K) = maximal G H.
Proof. by move=> nKG nKH; rewrite -cosetpre_maximal ?quotientGK. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
quotient_maximal
| |
quotient_maximal_eq:
K <| G -> K <| H -> maximal_eq (G / K) (H / K) = maximal_eq G H.
Proof. by move=> nKG nKH; rewrite -cosetpre_maximal_eq ?quotientGK. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
quotient_maximal_eq
| |
maximalJx : maximal (G :^ x) (H :^ x) = maximal G H.
Proof.
rewrite -{1}(setTI G) -{1}(setTI H) -!morphim_conj.
by rewrite injm_maximal ?subsetT ?injm_conj.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maximalJ
| |
maximal_eqJx : maximal_eq (G :^ x) (H :^ x) = maximal_eq G H.
Proof. by rewrite /maximal_eq !eqEsubset !conjSg maximalJ. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maximal_eqJ
| |
maxnormal_normalA B : maxnormal A B B -> A <| B.
Proof.
by case/maxsetP=> /and3P[/gen_set_id /= -> pAB nAB]; rewrite /normal proper_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maxnormal_normal
| |
maxnormal_properA B C : maxnormal A B C -> A \proper B.
Proof.
by case/maxsetP=> /and3P[gA pAB _] _; apply: (sub_proper_trans (subset_gen A)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maxnormal_proper
| |
maxnormal_subA B C : maxnormal A B C -> A \subset B.
Proof.
by move=> maxA; rewrite proper_sub //; apply: (maxnormal_proper maxA).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maxnormal_sub
| |
ex_maxnormal_ntrivgG : G :!=: 1-> {N : {group gT} | maxnormal N G G}.
Proof.
move=> ntG; apply: ex_maxgroup; exists [1 gT]%G; rewrite norm1 proper1G.
by rewrite subsetT ntG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
ex_maxnormal_ntrivg
| |
maxnormalMG H K :
maxnormal H G G -> maxnormal K G G -> H :<>: K -> H * K = G.
Proof.
move=> maxH maxK /eqP; apply: contraNeq => ltHK_G.
have [nsHG nsKG] := (maxnormal_normal maxH, maxnormal_normal maxK).
have cHK: commute H K.
exact: normC (subset_trans (normal_sub nsHG) (normal_norm nsKG)).
wlog suffices: H K {maxH} maxK nsHG nsKG cHK ltHK_G / H \subset K.
by move=> IH; rewrite eqEsubset !IH // -cHK.
have{maxK} /maxgroupP[_ maxK] := maxK.
apply/joing_idPr/maxK; rewrite ?joing_subr //= comm_joingE //.
by rewrite properEneq ltHK_G; apply: normalM.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maxnormalM
| |
maxnormal_minnormalG L M :
G \subset 'N(M) -> L \subset 'N(G) -> maxnormal M G L ->
minnormal (G / M) (L / M).
Proof.
move=> nMG nGL /maxgroupP[/andP[/andP[sMG ltMG] nML] maxM]; apply/mingroupP.
rewrite -subG1 quotient_sub1 ?ltMG ?quotient_norms //.
split=> // Hb /andP[ntHb nHbL]; have nsMG: M <| G by apply/andP.
case/inv_quotientS=> // H defHb sMH sHG; rewrite defHb; congr (_ / M).
apply/eqP; rewrite eqEproper sHG /=; apply: contra ntHb => ltHG.
have nsMH: M <| H := normalS sMH sHG nsMG.
rewrite defHb quotientS1 // (maxM H) // ltHG /= -(quotientGK nsMH) -defHb.
exact: norm_quotient_pre.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
maxnormal_minnormal
| |
minnormal_maxnormalG L M :
M <| G -> L \subset 'N(M) -> minnormal (G / M) (L / M) -> maxnormal M G L.
Proof.
case/andP=> sMG nMG nML /mingroupP[/andP[/= ntGM _] minGM]; apply/maxgroupP.
split=> [|H /andP[/andP[sHG ltHG] nHL] sMH].
by rewrite /proper sMG nML andbT; apply: contra ntGM => /quotientS1 ->.
apply/eqP; rewrite eqEsubset sMH andbT -quotient_sub1 ?(subset_trans sHG) //.
rewrite subG1; apply: contraR ltHG => ntHM; rewrite -(quotientSGK nMG) //.
by rewrite (minGM (H / M)%G) ?quotientS // ntHM quotient_norms.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
minnormal_maxnormal
| |
simplePgT (G : {group gT}) :
reflect (G :!=: 1 /\ forall H : {group gT}, H <| G -> H :=: 1 \/ H :=: G)
(simple G).
Proof.
apply: (iffP mingroupP); rewrite normG andbT => [[ntG simG]].
split=> // N /andP[sNG nNG].
by case: (eqsVneq N 1) => [|ntN]; [left | right; apply: simG; rewrite ?ntN].
split=> // N /andP[ntN nNG] sNG.
by case: (simG N) ntN => // [|->]; [apply/andP | case/eqP].
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
simpleP
| |
quotient_simplegT (G H : {group gT}) :
H <| G -> simple (G / H) = maxnormal H G G.
Proof.
move=> nsHG; have nGH := normal_norm nsHG.
by apply/idP/idP; [apply: minnormal_maxnormal | apply: maxnormal_minnormal].
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
quotient_simple
| |
isog_simplegT rT (G : {group gT}) (M : {group rT}) :
G \isog M -> simple G = simple M.
Proof.
move=> eqGM; wlog suffices: gT rT G M eqGM / simple M -> simple G.
by move=> IH; apply/idP/idP; apply: IH; rewrite // isog_sym.
case/isogP: eqGM => f injf <- /simpleP[ntGf simGf].
apply/simpleP; split=> [|N nsNG]; first by rewrite -(morphim_injm_eq1 injf).
rewrite -(morphim_invm injf (normal_sub nsNG)).
have: f @* N <| f @* G by rewrite morphim_normal.
by case/simGf=> /= ->; [left | right]; rewrite (morphim1, morphim_invm).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
isog_simple
| |
simple_maxnormalgT (G : {group gT}) : simple G = maxnormal 1 G G.
Proof.
by rewrite -quotient_simple ?normal1 // -(isog_simple (quotient1_isog G)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
simple_maxnormal
| |
chief_factor_minnormalG V U :
chief_factor G V U -> minnormal (U / V) (G / V).
Proof.
case/andP=> maxV /andP[sUG nUG]; apply: maxnormal_minnormal => //.
by have /andP[_ nVG] := maxgroupp maxV; apply: subset_trans sUG nVG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
chief_factor_minnormal
| |
acts_irrQG U V :
G \subset 'N(V) -> V <| U ->
acts_irreducibly G (U / V) 'Q = minnormal (U / V) (G / V).
Proof.
move=> nVG nsVU; apply/mingroupP/mingroupP; case=> /andP[->] /=.
rewrite astabsQ // subsetI nVG /= => nUG minUV.
rewrite quotient_norms //; split=> // H /andP[ntH nHG] sHU.
by apply: minUV (sHU); rewrite ntH -(cosetpreK H) actsQ // norm_quotient_pre.
rewrite sub_quotient_pre // => nUG minU; rewrite astabsQ //.
rewrite (subset_trans nUG); last first.
by rewrite subsetI subsetIl /= -{2}(quotientGK nsVU) morphpre_norm.
split=> // H /andP[ntH nHG] sHU.
rewrite -{1}(cosetpreK H) astabsQ ?normal_cosetpre ?subsetI ?nVG //= in nHG.
apply: minU sHU; rewrite ntH; apply: subset_trans (quotientS _ nHG) _.
by rewrite -{2}(cosetpreK H) quotient_norm.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
acts_irrQ
| |
chief_series_existsH G :
H <| G -> {s | (G.-chief).-series 1%G s & last 1%G s = H}.
Proof.
have [m] := ubnP #|H|; elim: m H => // m IHm U leUm nsUG.
have [-> | ntU] := eqVneq U 1%G; first by exists [::].
have [V maxV]: {V : {group gT} | maxnormal V U G}.
by apply: ex_maxgroup; exists 1%G; rewrite proper1G ntU norms1.
have /andP[ltVU nVG] := maxgroupp maxV.
have [||s ch_s defV] := IHm V; first exact: leq_trans (proper_card ltVU) _.
by rewrite /normal (subset_trans (proper_sub ltVU) (normal_sub nsUG)).
exists (rcons s U); last by rewrite last_rcons.
by rewrite rcons_path defV /= ch_s /chief_factor; apply/and3P.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
chief_series_exists
| |
central_factor_centralH K :
central_factor G H K -> (K / H) \subset 'Z(G / H).
Proof. by case/and3P=> /quotient_cents2r *; rewrite subsetI quotientS. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
central_factor_central
| |
central_central_factorH K :
(K / H) \subset 'Z(G / H) -> H <| K -> H <| G -> central_factor G H K.
Proof.
case/subsetIP=> sKGb cGKb /andP[sHK nHK] /andP[sHG nHG].
by rewrite /central_factor -quotient_cents2 // cGKb sHK -(quotientSGK nHK).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action commutator center"
] |
solvable/gseries.v
|
central_central_factor
| |
SchurZassenhaus_splitgT (G H : {group gT}) :
Hall G H -> H <| G -> [splits G, over H].
Proof.
have [n] := ubnP #|G|; elim: n => // n IHn in gT G H * => /ltnSE-Gn hallH nsHG.
have [sHG nHG] := andP nsHG.
have [-> | [p pr_p pH]] := trivgVpdiv H.
by apply/splitsP; exists G; rewrite inE -subG1 subsetIl mul1g eqxx.
have [P sylP] := Sylow_exists p H.
case nPG: (P <| G); last first.
pose N := ('N_G(P))%G; have sNG: N \subset G by rewrite subsetIl.
have eqHN_G: H * N = G by apply: Frattini_arg sylP.
pose H' := (H :&: N)%G.
have nsH'N: H' <| N.
by rewrite /normal subsetIr normsI ?normG ?(subset_trans sNG).
have eq_iH: #|G : H| = #|N| %/ #|H'|.
rewrite -divgS // -(divnMl (cardG_gt0 H')) mulnC -eqHN_G.
by rewrite -mul_cardG (mulnC #|H'|) divnMl // cardG_gt0.
have hallH': Hall N H'.
rewrite /Hall -divgS subsetIr //= -eq_iH.
by case/andP: hallH => _; apply: coprimeSg; apply: subsetIl.
have: [splits N, over H'].
apply: IHn hallH' nsH'N; apply: {n}leq_trans Gn.
rewrite proper_card // properEneq sNG andbT; apply/eqP=> eqNG.
by rewrite -eqNG normal_subnorm (subset_trans (pHall_sub sylP)) in nPG.
case/splitsP=> K /complP[tiKN eqH'K].
have sKN: K \subset N by rewrite -(mul1g K) -eqH'K mulSg ?sub1set.
apply/splitsP; exists K; rewrite inE -subG1; apply/andP; split.
by rewrite /= -(setIidPr sKN) setIA tiKN.
by rewrite eqEsubset -eqHN_G mulgS // -eqH'K mulGS mulSg ?subsetIl.
pose Z := 'Z(P); pose Gbar := G / Z; pose Hbar := H / Z.
have sZP: Z \subset P by ap
...
|
Theorem
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
SchurZassenhaus_split
| |
SchurZassenhaus_trans_solgT (H K K1 : {group gT}) :
solvable H -> K \subset 'N(H) -> K1 \subset H * K ->
coprime #|H| #|K| -> #|K1| = #|K| ->
exists2 x, x \in H & K1 :=: K :^ x.
Proof.
have [n] := ubnP #|H|.
elim: n => // n IHn in gT H K K1 * => /ltnSE-leHn solH nHK.
have [-> | ] := eqsVneq H 1.
rewrite mul1g => sK1K _ eqK1K; exists 1; first exact: set11.
by apply/eqP; rewrite conjsg1 eqEcard sK1K eqK1K /=.
pose G := (H <*> K)%G.
have defG: G :=: H * K by rewrite -normC // -norm_joinEl // joingC.
have sHG: H \subset G by apply: joing_subl.
have sKG: K \subset G by apply: joing_subr.
have nsHG: H <| G by rewrite /(H <| G) sHG join_subG normG.
case/(solvable_norm_abelem solH nsHG)=> M [sMH nsMG ntM] /and3P[_ abelM _].
have [sMG nMG] := andP nsMG; rewrite -defG => sK1G coHK oK1K.
have nMsG (L : {set gT}): L \subset G -> L \subset 'N(M).
by move/subset_trans->.
have [coKM coHMK]: coprime #|M| #|K| /\ coprime #|H / M| #|K|.
by apply/andP; rewrite -coprimeMl card_quotient ?nMsG ?Lagrange.
have oKM (K' : {group gT}): K' \subset G -> #|K'| = #|K| -> #|K' / M| = #|K|.
move=> sK'G oK'.
rewrite -quotientMidr -?norm_joinEl ?card_quotient ?nMsG //; last first.
by rewrite gen_subG subUset sK'G.
rewrite -divgS /=; last by rewrite -gen_subG genS ?subsetUr.
by rewrite norm_joinEl ?nMsG // coprime_cardMg ?mulnK // oK' coprime_sym.
have [xb]: exists2 xb, xb \in H / M & K1 / M = (K / M) :^ xb.
apply: IHn; try by rewrite (quotient_sol, morphim_norms, oKM K) ?(oKM K1).
by apply: leq_tra
...
|
Theorem
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
SchurZassenhaus_trans_sol
| |
SchurZassenhaus_trans_actsolgT (G A B : {group gT}) :
solvable A -> A \subset 'N(G) -> B \subset A <*> G ->
coprime #|G| #|A| -> #|A| = #|B| ->
exists2 x, x \in G & B :=: A :^ x.
Proof.
set AG := A <*> G; have [n] := ubnP #|AG|.
elim: n => // n IHn in gT A B G AG * => /ltnSE-leAn solA nGA sB_AG coGA oAB.
have [A1 | ntA] := eqsVneq A 1.
by exists 1; rewrite // conjsg1 A1 (@card1_trivg _ B) // -oAB A1 cards1.
have [M [sMA nsMA ntM]] := solvable_norm_abelem solA (normal_refl A) ntA.
case/is_abelemP=> q q_pr /abelem_pgroup qM; have nMA := normal_norm nsMA.
have defAG: AG = A * G := norm_joinEl nGA.
have sA_AG: A \subset AG := joing_subl _ _.
have sG_AG: G \subset AG := joing_subr _ _.
have sM_AG := subset_trans sMA sA_AG.
have oAG: #|AG| = (#|A| * #|G|)%N by rewrite defAG coprime_cardMg 1?coprime_sym.
have q'G: (#|G|`_q = 1)%N.
rewrite part_p'nat ?p'natE -?prime_coprime // coprime_sym.
have [_ _ [k oM]] := pgroup_pdiv qM ntM.
by rewrite -(@coprime_pexpr k.+1) // -oM (coprimegS sMA).
have coBG: coprime #|B| #|G| by rewrite -oAB coprime_sym.
have defBG: B * G = AG.
by apply/eqP; rewrite eqEcard mul_subG ?sG_AG //= oAG oAB coprime_cardMg.
case nMG: (G \subset 'N(M)).
have nsM_AG: M <| AG by rewrite /normal sM_AG join_subG nMA.
have nMB: B \subset 'N(M) := subset_trans sB_AG (normal_norm nsM_AG).
have sMB: M \subset B.
have [Q sylQ]:= Sylow_exists q B; have sQB := pHall_sub sylQ.
apply: subset_trans (normal_sub_max_pgroup (Hall_max _) qM nsM_AG) (sQB).
rewrite pHallE (subse
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
SchurZassenhaus_trans_actsol
| |
Hall_exists_subJpi gT (G : {group gT}) :
solvable G -> exists2 H : {group gT}, pi.-Hall(G) H
& forall K : {group gT}, K \subset G -> pi.-group K ->
exists2 x, x \in G & K \subset H :^ x.
Proof.
have [n] := ubnP #|G|; elim: n gT G => // n IHn gT G /ltnSE-leGn solG.
have [-> | ntG] := eqsVneq G 1.
exists 1%G => [|_ /trivGP-> _]; last by exists 1; rewrite ?set11 ?sub1G.
by rewrite pHallE sub1G cards1 part_p'nat.
case: (solvable_norm_abelem solG (normal_refl _)) => // M [sMG nsMG ntM].
case/is_abelemP=> p pr_p /and3P[pM cMM _].
pose Gb := (G / M)%G; case: (IHn _ Gb) => [||Hb]; try exact: quotient_sol.
by rewrite (leq_trans (ltn_quotient _ _)).
case/and3P=> [sHbGb piHb pi'Hb'] transHb.
case: (inv_quotientS nsMG sHbGb) => H def_H sMH sHG.
have nMG := normal_norm nsMG; have nMH := subset_trans sHG nMG.
have{transHb} transH (K : {group gT}):
K \subset G -> pi.-group K -> exists2 x, x \in G & K \subset H :^ x.
- move=> sKG piK; have nMK := subset_trans sKG nMG.
case: (transHb (K / M)%G) => [||xb Gxb sKHxb]; first exact: morphimS.
exact: morphim_pgroup.
case/morphimP: Gxb => x Nx Gx /= def_x; exists x => //.
apply/subsetP=> y Ky.
have: y \in coset M y by rewrite val_coset (subsetP nMK, rcoset_refl).
have: coset M y \in (H :^ x) / M.
rewrite /quotient morphimJ //=.
by rewrite def_x def_H in sKHxb; apply/(subsetP sKHxb)/mem_quotient.
case/morphimP=> z Nz Hxz ->.
rewrite val_coset //; case/rcosetP=> t Mt ->; rewrite groupMl //.
by re
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
Hall_exists_subJ
| |
Hall_Frattini_argpi (G K H : {group gT}) :
solvable K -> K <| G -> pi.-Hall(K) H -> K * 'N_G(H) = G.
Proof.
move=> solK /andP[sKG nKG] hallH.
have sHG: H \subset G by apply: subset_trans sKG; case/andP: hallH.
rewrite setIC group_modl //; apply/setIidPr/subsetP=> x Gx.
pose H1 := (H :^ x^-1)%G.
have hallH1: pi.-Hall(K) H1 by rewrite pHallJnorm // groupV (subsetP nKG).
case: (Hall_trans solK hallH hallH1) => y Ky defH.
rewrite -(mulKVg y x) mem_mulg //; apply/normP.
by rewrite conjsgM {1}defH conjsgK conjsgKV.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
Hall_Frattini_arg
| |
coprime_norm_centA G :
A \subset 'N(G) -> coprime #|G| #|A| -> 'N_G(A) = 'C_G(A).
Proof.
move=> nGA coGA; apply/eqP; rewrite eqEsubset andbC setIS ?cent_sub //=.
rewrite subsetI subsetIl /= (sameP commG1P trivgP) -(coprime_TIg coGA).
rewrite subsetI commg_subr subsetIr andbT.
move: nGA; rewrite -commg_subl; apply: subset_trans.
by rewrite commSg ?subsetIl.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent",
"From mathcomp Require Import sylow abelian maximal"
] |
solvable/hall.v
|
coprime_norm_cent
|
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