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 ... | 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 sF... | 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 morphi... | 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 // ... | 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 \s... | 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.
cas... | 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=> //... | 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 max... | 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.
spli... | 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_m... | 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; rewr... | 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 {ma... | 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... | 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; re... | 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 /... | 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... | 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); ... | 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 norms... | 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... | 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... | 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 mul... | 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... | 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... | 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... | 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] := ... | 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... | 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(... | 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... | 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: su... | 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... | solvable/hall.v | coprime_norm_cent |
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