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 |
|---|---|---|---|---|---|---|
subcent_charG H K : H \char G -> K \char G -> 'C_H(K) \char G.
Proof.
case/charP=> sHG chHG /charP[sKG chKG]; apply/charP.
split=> [|f injf Gf]; first by rewrite subIset ?sHG.
by rewrite injm_subcent ?chHG ?chKG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
subcent_char
| |
centerPA x : reflect (x \in A /\ centralises x A) (x \in 'Z(A)).
Proof. exact: subcentP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
centerP
| |
center_subA : 'Z(A) \subset A.
Proof. exact: subsetIl. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_sub
| |
center1: 'Z(1) = 1 :> {set gT}.
Proof. exact: gF1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center1
| |
centerCA : {in A, centralised 'Z(A)}.
Proof. by apply/centsP; rewrite centsC subsetIr. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
centerC
| |
center_normalG : 'Z(G) <| G.
Proof. exact: gFnormal. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_normal
| |
sub_center_normalH G : H \subset 'Z(G) -> H <| G.
Proof. by rewrite subsetI centsC /normal => /andP[-> /cents_norm]. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
sub_center_normal
| |
center_abelianG : abelian 'Z(G).
Proof. by rewrite /abelian subIset // centsC subIset // subxx orbT. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_abelian
| |
center_charG : 'Z(G) \char G.
Proof. exact: gFchar. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_char
| |
center_idPA : reflect ('Z(A) = A) (abelian A).
Proof. exact: setIidPl. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_idP
| |
center_class_formulaG :
#|G| = #|'Z(G)| + \sum_(xG in [set x ^: G | x in G :\: 'C(G)]) #|xG|.
Proof.
by rewrite acts_sum_card_orbit ?cardsID // astabsJ normsD ?norms_cent ?normG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_class_formula
| |
subcent1PA x y : reflect (y \in A /\ commute x y) (y \in 'C_A[x]).
Proof.
rewrite inE; case: (y \in A); last by right; case.
by apply: (iffP cent1P) => [|[]].
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
subcent1P
| |
subcent1_idx G : x \in G -> x \in 'C_G[x].
Proof. by move=> Gx; rewrite inE Gx; apply/cent1P. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
subcent1_id
| |
subcent1_subx G : 'C_G[x] \subset G.
Proof. exact: subsetIl. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
subcent1_sub
| |
subcent1Cx y G : x \in G -> y \in 'C_G[x] -> x \in 'C_G[y].
Proof. by move=> Gx /subcent1P[_ cxy]; apply/subcent1P. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
subcent1C
| |
subcent1_cycle_subx G : x \in G -> <[x]> \subset 'C_G[x].
Proof. by move=> Gx; rewrite cycle_subG ?subcent1_id. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
subcent1_cycle_sub
| |
subcent1_cycle_normx G : 'C_G[x] \subset 'N(<[x]>).
Proof. by rewrite cents_norm // cent_gen cent_set1 subsetIr. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
subcent1_cycle_norm
| |
subcent1_cycle_normalx G : x \in G -> <[x]> <| 'C_G[x].
Proof.
by move=> Gx; rewrite /normal subcent1_cycle_norm subcent1_cycle_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
subcent1_cycle_normal
| |
cyclic_center_factor_abelianG : cyclic (G / 'Z(G)) -> abelian G.
Proof.
case/cyclicP=> a Ga; case: (cosetP a) => /= z Nz def_a.
have G_Zz: G :=: 'Z(G) * <[z]>.
rewrite -quotientK ?cycle_subG ?quotient_cycle //=.
by rewrite -def_a -Ga quotientGK // center_normal.
rewrite G_Zz abelianM cycle_abelian center_abelian centsC /= G_Zz.
by rewrite subIset ?centS ?orbT ?mulG_subr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cyclic_center_factor_abelian
| |
cyclic_factor_abelianH G :
H \subset 'Z(G) -> cyclic (G / H) -> abelian G.
Proof.
move=> sHZ cycGH; apply: cyclic_center_factor_abelian.
have /andP[_ nHG]: H <| G := sub_center_normal sHZ.
have [f <-]:= homgP (homg_quotientS nHG (gFnorm _ G) sHZ).
exact: morphim_cyclic cycGH.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cyclic_factor_abelian
| |
injm_centerG : G \subset D -> f @* 'Z(G) = 'Z(f @* G).
Proof. exact: injm_subcent. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
injm_center
| |
isog_center(aT rT : finGroupType) (G : {group aT}) (H : {group rT}) :
G \isog H -> 'Z(G) \isog 'Z(H).
Proof. exact: gFisog. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
isog_center
| |
center_prodH K : K \subset 'C(H) -> 'Z(H) * 'Z(K) = 'Z(H * K).
Proof.
move=> cHK; apply/setP=> z; rewrite {3}/center centM !inE.
have cKH: H \subset 'C(K) by rewrite centsC.
apply/imset2P/and3P=> [[x y /setIP[Hx cHx] /setIP[Ky cKy] ->{z}]| []].
by rewrite imset2_f ?groupM // ?(subsetP cHK) ?(subsetP cKH).
case/imset2P=> x y Hx Ky ->{z}.
rewrite groupMr => [cHx|]; last exact: subsetP Ky.
rewrite groupMl => [cKy|]; last exact: subsetP Hx.
by exists x y; rewrite ?inE ?Hx ?Ky.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_prod
| |
center_cprodA B G : A \* B = G -> 'Z(A) \* 'Z(B) = 'Z(G).
Proof.
case/cprodP => [[H K -> ->] <- cHK].
rewrite cprodE ?center_prod //= subIset ?(subset_trans cHK) //.
by rewrite centS ?center_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_cprod
| |
center_bigcprodI r P (F : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) F i = G ->
\big[cprod/1]_(i <- r | P i) 'Z(F i) = 'Z(G).
Proof.
elim/big_ind2: _ G => [_ <-|A B C D IHA IHB G dG|_ _ G ->]; rewrite ?center1 //.
case/cprodP: dG IHA IHB (dG) => [[H K -> ->] _ _] IHH IHK dG.
by rewrite (IHH H) // (IHK K) // (center_cprod dG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_bigcprod
| |
cprod_center_idG : G \* 'Z(G) = G.
Proof. by rewrite cprodE ?subsetIr // mulGSid ?center_sub. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cprod_center_id
| |
center_dprodA B G : A \x B = G -> 'Z(A) \x 'Z(B) = 'Z(G).
Proof.
case/dprodP=> [[H1 H2 -> ->] defG cH12 trH12].
move: defG; rewrite -cprodE // => /center_cprod/cprodP[_ /= <- cZ12].
by apply: dprodE; rewrite //= setIAC setIA -setIA trH12 (setIidPl _) ?sub1G.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_dprod
| |
center_bigdprodI r P (F: I -> {set gT}) G :
\big[dprod/1]_(i <- r | P i) F i = G ->
\big[dprod/1]_(i <- r | P i) 'Z(F i) = 'Z(G).
Proof.
elim/big_ind2: _ G => [_ <-|A B C D IHA IHB G dG|_ _ G ->]; rewrite ?center1 //.
case/dprodP: dG IHA IHB (dG) => [[H K -> ->] _ _ _] IHH IHK dG.
by rewrite (IHH H) // (IHK K) // (center_dprod dG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_bigdprod
| |
Aut_cprod_fullG H K :
H \* K = G -> 'Z(H) = 'Z(K) ->
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
Aut_in (Aut K) 'Z(K) \isog Aut 'Z(K) ->
Aut_in (Aut G) 'Z(G) \isog Aut 'Z(G).
Proof.
move=> defG eqZHK; have [_ defHK cHK] := cprodP defG.
have defZ: 'Z(G) = 'Z(H) by rewrite -defHK -center_prod // eqZHK mulGid.
have ziHK: H :&: K = 'Z(K).
by apply/eqP; rewrite eqEsubset subsetI -{1 2}eqZHK !center_sub setIS.
have AutZP := Aut_sub_fullP (@center_sub gT _).
move/AutZP=> AutZHfull /AutZP AutZKfull; apply/AutZP=> g injg gZ.
have [gH [def_gH ker_gH _ im_gH]] := domP g defZ.
have [gK [def_gK ker_gK _ im_gK]] := domP g (etrans defZ eqZHK).
have [injgH injgK]: 'injm gH /\ 'injm gK by rewrite ker_gH ker_gK.
have [gHH gKK]: gH @* 'Z(H) = 'Z(H) /\ gK @* 'Z(K) = 'Z(K).
by rewrite im_gH im_gK -eqZHK -defZ.
have [|fH [injfH im_fH fHZ]] := AutZHfull gH injgH.
by rewrite im_gH /= -defZ.
have [|fK [injfK im_fK fKZ]] := AutZKfull gK injgK.
by rewrite im_gK /= -eqZHK -defZ.
have cfHK: fK @* K \subset 'C(fH @* H) by rewrite im_fH im_fK.
have eq_fHK: {in H :&: K, fH =1 fK}.
by move=> z; rewrite ziHK => Zz; rewrite fHZ ?fKZ /= ?eqZHK // def_gH def_gK.
exists (cprodm_morphism defG cfHK eq_fHK).
rewrite injm_cprodm injfH injfK im_cprodm im_fH im_fK defHK.
rewrite -morphimIdom ziHK -eqZHK injm_center // im_fH eqxx.
split=> //= z; rewrite {1}defZ => Zz; have [Hz _] := setIP Zz.
by rewrite cprodmEl // fHZ // def_gH.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
Aut_cprod_full
| |
ker_cprod_byof isom 'Z(H) 'Z(K) gz :=
[set xy | let: (x, y) := xy in (x \in 'Z(H)) && (y == (gz x)^-1)].
Hypothesis isoZ : isom 'Z(H) 'Z(K) gz.
Let kerHK := ker_cprod_by isoZ.
Let injgz : 'injm gz. Proof. by case/isomP: isoZ. Qed.
Let gzZ : gz @* 'Z(H) = 'Z(K). Proof. by case/isomP: isoZ. Qed.
Let gzZchar : gz @* 'Z(H) \char 'Z(K). Proof. by rewrite gzZ. Qed.
Let sgzZZ : gz @* 'Z(H) \subset 'Z(K) := char_sub gzZchar.
Let sZH := center_sub H.
Let sZK := center_sub K.
Let sgzZG : gz @* 'Z(H) \subset K := subset_trans sgzZZ sZK.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ker_cprod_by
| |
ker_cprod_by_is_group: group_set kerHK.
Proof.
apply/group_setP; rewrite inE /= group1 morph1 invg1 /=.
split=> // [[x1 y1] [x2 y2]].
rewrite inE /= => /andP[Zx1 /eqP->]; have [_ cGx1] := setIP Zx1.
rewrite inE /= => /andP[Zx2 /eqP->]; have [Gx2 _] := setIP Zx2.
by rewrite inE /= groupM //= -invMg (centP cGx1) // morphM.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ker_cprod_by_is_group
| |
ker_cprod_by_group:= Group ker_cprod_by_is_group.
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ker_cprod_by_group
| |
ker_cprod_by_central: kerHK \subset 'Z(setX H K).
Proof.
rewrite -(center_dprod (setX_dprod H K)) -morphim_pairg1 -morphim_pair1g.
rewrite -!injm_center ?subsetT ?injm_pair1g ?injm_pairg1 //=.
rewrite morphim_pairg1 morphim_pair1g setX_dprod.
apply/subsetP=> [[x y]] /[1!inE] /andP[Zx /eqP->].
by rewrite inE /= Zx groupV (subsetP sgzZZ) ?mem_morphim.
Qed.
Fact cprod_by_key : unit. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ker_cprod_by_central
| |
cprod_by_def: finGroupType := subg_of (setX H K / kerHK).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cprod_by_def
| |
cprod_by:= locked_with cprod_by_key cprod_by_def.
Local Notation C := [set: FinGroup.sort cprod_by].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cprod_by
| |
in_cprod: gTH * gTK -> cprod_by :=
let: tt as k := cprod_by_key return _ -> locked_with k cprod_by_def in
subg _ \o coset kerHK.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
in_cprod
| |
in_cprodM: {in setX H K &, {morph in_cprod : u v / u * v}}.
Proof.
rewrite /in_cprod /cprod_by; case: cprod_by_key => /= u v Gu Gv.
have nkerHKG := normal_norm (sub_center_normal ker_cprod_by_central).
by rewrite -!morphM ?mem_quotient // (subsetP nkerHKG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
in_cprodM
| |
in_cprod_morphism:= Morphism in_cprodM.
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
in_cprod_morphism
| |
ker_in_cprod: 'ker in_cprod = kerHK.
Proof.
transitivity ('ker (subg [group of setX H K / kerHK] \o coset kerHK)).
rewrite /ker /morphpre /= /in_cprod /cprod_by; case: cprod_by_key => /=.
by rewrite ['N(_) :&: _]quotientGK ?sub_center_normal ?ker_cprod_by_central.
by rewrite ker_comp ker_subg -kerE ker_coset.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ker_in_cprod
| |
cpairg1_dom: H \subset 'dom (in_cprod \o @pairg1 gTH gTK).
Proof. by rewrite -sub_morphim_pre ?subsetT // morphim_pairg1 setXS ?sub1G. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cpairg1_dom
| |
cpair1g_dom: K \subset 'dom (in_cprod \o @pair1g gTH gTK).
Proof. by rewrite -sub_morphim_pre ?subsetT // morphim_pair1g setXS ?sub1G. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cpair1g_dom
| |
cpairg1:= tag (restrmP _ cpairg1_dom).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cpairg1
| |
cpair1g:= tag (restrmP _ cpair1g_dom).
Local Notation CH := (mfun cpairg1 @* gval H).
Local Notation CK := (mfun cpair1g @* gval K).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cpair1g
| |
injm_cpairg1: 'injm cpairg1.
Proof.
rewrite /cpairg1; case: restrmP => _ [_ -> _ _].
rewrite ker_comp ker_in_cprod; apply/subsetP=> x; rewrite !inE /=.
by case/and3P=> _ Zx; rewrite eq_sym (inv_eq invgK) invg1 morph_injm_eq1.
Qed.
Let injH := injm_cpairg1.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
injm_cpairg1
| |
injm_cpair1g: 'injm cpair1g.
Proof.
rewrite /cpair1g; case: restrmP => _ [_ -> _ _].
rewrite ker_comp ker_in_cprod; apply/subsetP=> y; rewrite !inE /= morph1 invg1.
by case/and3P.
Qed.
Let injK := injm_cpair1g.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
injm_cpair1g
| |
im_cpair_cent: CK \subset 'C(CH).
Proof.
rewrite /cpairg1 /cpair1g; do 2!case: restrmP => _ [_ _ _ -> //].
rewrite !morphim_comp morphim_cents // morphim_pair1g morphim_pairg1.
by case/dprodP: (setX_dprod H K).
Qed.
Hint Resolve im_cpair_cent : core.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
im_cpair_cent
| |
im_cpair: CH * CK = C.
Proof.
rewrite /cpairg1 /cpair1g; do 2!case: restrmP => _ [_ _ _ -> //].
rewrite !morphim_comp -morphimMl morphim_pairg1 ?setXS ?sub1G //.
rewrite morphim_pair1g setX_prod morphimEdom /= /in_cprod /cprod_by.
by case: cprod_by_key; rewrite /= imset_comp imset_coset -morphimEdom im_subg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
im_cpair
| |
im_cpair_cprod: CH \* CK = C. Proof. by rewrite cprodE ?im_cpair. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
im_cpair_cprod
| |
eq_cpairZ: {in 'Z(H), cpairg1 =1 cpair1g \o gz}.
Proof.
rewrite /cpairg1 /cpair1g => z1 Zz1; set z2 := gz z1.
have Zz2: z2 \in 'Z(K) by rewrite (subsetP sgzZZ) ?mem_morphim.
have [[Gz1 _] [/= Gz2 _]]:= (setIP Zz1, setIP Zz2).
do 2![case: restrmP => f /= [df _ _ _]; rewrite {f}df].
apply/rcoset_kerP; rewrite ?inE ?group1 ?andbT //.
by rewrite ker_in_cprod mem_rcoset inE /= invg1 mulg1 mul1g Zz1 /=.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
eq_cpairZ
| |
setI_im_cpair: CH :&: CK = 'Z(CH).
Proof.
apply/eqP; rewrite eqEsubset setIS //=.
rewrite subsetI center_sub -injm_center //.
rewrite (eq_in_morphim _ eq_cpairZ); first by rewrite morphim_comp morphimS.
by rewrite !(setIidPr _) // -sub_morphim_pre.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
setI_im_cpair
| |
cpair1g_center: cpair1g @* 'Z(K) = 'Z(C).
Proof.
case/cprodP: (center_cprod im_cpair_cprod) => _ <- _.
by rewrite injm_center // -setI_im_cpair mulSGid //= setIC setIS 1?centsC.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cpair1g_center
| |
cpair_center_id: 'Z(CH) = 'Z(CK).
Proof.
rewrite -!injm_center // -gzZ -morphim_comp; apply: eq_in_morphim eq_cpairZ.
by rewrite !(setIidPr _) // -sub_morphim_pre.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cpair_center_id
| |
cpairg1_center: cpairg1 @* 'Z(H) = 'Z(C).
Proof. by rewrite -cpair1g_center !injm_center // cpair_center_id. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cpairg1_center
| |
xcprodm_cent: gK @* CK \subset 'C(gH @* CH).
Proof. by rewrite !im_ifactm. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprodm_cent
| |
xcprodmI: {in CH :&: CK, gH =1 gK}.
Proof.
rewrite setI_im_cpair -injm_center // => fHx; case/morphimP=> x Gx Zx ->{fHx}.
by rewrite {2}eq_cpairZ //= ?ifactmE ?eq_fHK //= (subsetP sgzZG) ?mem_morphim.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprodmI
| |
xcprodm:= cprodm im_cpair_cprod xcprodm_cent xcprodmI.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprodm
| |
xcprod_morphism:= [morphism of xcprodm].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprod_morphism
| |
xcprodmEl: {in H, forall x, xcprodm (cpairg1 x) = fH x}.
Proof. by move=> x Hx; rewrite /xcprodm cprodmEl ?mem_morphim ?ifactmE. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprodmEl
| |
xcprodmEr: {in K, forall y, xcprodm (cpair1g y) = fK y}.
Proof. by move=> y Ky; rewrite /xcprodm cprodmEr ?mem_morphim ?ifactmE. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprodmEr
| |
xcprodmE:
{in H & K, forall x y, xcprodm (cpairg1 x * cpair1g y) = fH x * fK y}.
Proof.
by move=> x y Hx Ky; rewrite /xcprodm cprodmE ?mem_morphim ?ifactmE.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprodmE
| |
im_xcprodm: xcprodm @* C = fH @* H * fK @* K.
Proof. by rewrite -im_cpair morphim_cprodm // !im_ifactm. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
im_xcprodm
| |
im_xcprodmlA : xcprodm @* (cpairg1 @* A) = fH @* A.
Proof.
rewrite -!(morphimIdom _ A) morphim_cprodml ?morphimS ?subsetIl //.
by rewrite morphim_ifactm ?subsetIl.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
im_xcprodml
| |
im_xcprodmrA : xcprodm @* (cpair1g @* A) = fK @* A.
Proof.
rewrite -!(morphimIdom _ A) morphim_cprodmr ?morphimS ?subsetIl //.
by rewrite morphim_ifactm ?subsetIl.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
im_xcprodmr
| |
injm_xcprodm: 'injm xcprodm = 'injm fH && 'injm fK.
Proof.
rewrite injm_cprodm !ker_ifactm !subG1 !morphim_injm_eq1 ?subsetIl // -!subG1.
apply: andb_id2l => /= injfH; apply: andb_idr => _.
rewrite !im_ifactm // -(morphimIdom gH) setI_im_cpair -injm_center //.
rewrite morphim_ifactm // eqEsubset subsetI morphimS //=.
rewrite {1}injm_center // setIS //=.
rewrite (eq_in_morphim _ eq_fHK); first by rewrite morphim_comp morphimS.
by rewrite !(setIidPr _) // -sub_morphim_pre.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
injm_xcprodm
| |
Aut_cprod_by_full:
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
Aut_in (Aut K) 'Z(K) \isog Aut 'Z(K) ->
Aut_in (Aut C) 'Z(C) \isog Aut 'Z(C).
Proof.
move=> AutZinH AutZinK.
have Cfull:= Aut_cprod_full im_cpair_cprod cpair_center_id.
by rewrite Cfull // -injm_center // injm_Aut_full ?center_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
Aut_cprod_by_full
| |
cprod_by_uniq:
exists f : {morphism G >-> cprod_by},
[/\ isom G C f, f @* GH = CH & f @* GK = CK].
Proof.
have [_ defGHK cGKH] := cprodP defG.
have AutZinH := Aut_sub_fullP sZH AutZHfull.
have [fH injfH defGH]:= isogP (isog_symr isoGH).
have [fK injfK defGK]:= isogP (isog_symr isoGK).
have sfHZfK: fH @* 'Z(H) \subset fK @* K.
by rewrite injm_center //= defGH defGK -ziGHK subsetIr.
have gzZ_id: gz @* 'Z(H) = invm injfK @* (fH @* 'Z(H)).
apply: gzZ_lone => /=.
rewrite injm_center // defGH -ziGHK sub_morphim_pre /= ?defGK ?subsetIr //.
by rewrite setIC morphpre_invm injm_center // defGK setIS 1?centsC.
rewrite -morphim_comp.
apply: isog_trans (sub_isog _ _); first by rewrite isog_sym sub_isog.
by rewrite -sub_morphim_pre.
by rewrite !injm_comp ?injm_invm.
have: 'dom (invm injfH \o fK \o gz) = 'Z(H).
rewrite /dom /= -(morphpreIdom gz); apply/setIidPl.
by rewrite -2?sub_morphim_pre // gzZ_id morphim_invmE morphpreK ?morphimS.
case/domP=> gzH [def_gzH ker_gzH _ im_gzH].
have{ker_gzH} injgzH: 'injm gzH by rewrite ker_gzH !injm_comp ?injm_invm.
have{AutZinH} [|gH [injgH gH_H def_gH]] := AutZinH _ injgzH.
by rewrite im_gzH !morphim_comp /= gzZ_id !morphim_invmE morphpreK ?injmK.
have: 'dom (fH \o gH) = H by rewrite /dom /= -{3}gH_H injmK.
case/domP=> gfH [def_gfH ker_gfH _ im_gfH].
have{im_gfH} gfH_H: gfH @* H = GH by rewrite im_gfH morphim_comp gH_H.
have cgfHfK: fK @* K \subset 'C(gfH @* H) by rewrite gfH_H defGK.
have eq_gfHK: {in 'Z(H), gfH =1 fK \o gz}.
mov
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
cprod_by_uniq
| |
isog_cprod_by: G \isog C.
Proof. by have [f [isoG _ _]] := cprod_by_uniq; apply: isom_isog isoG. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
isog_cprod_by
| |
xcprod_subproof:
{gz : {morphism 'Z(H) >-> gt_ isob} | isom 'Z(H) 'Z(G_ isob) gz}.
Proof.
case: (pickP [pred f : {ffun _} | misom 'Z(H) 'Z(K) f]) => [f isoZ | no_f].
rewrite (misom_isog isoZ); case/andP: isoZ => fM isoZ.
by exists [morphism of morphm fM].
move/pred0P: no_f => not_isoZ; rewrite [isob](congr1 negb not_isoZ).
by exists (idm_morphism _); apply/isomP; rewrite injm_idm im_idm.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprod_subproof
| |
xcprod:= cprod_by (svalP xcprod_subproof).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprod
| |
xcprod_spec: finGroupType -> Prop :=
XcprodSpec gz isoZ : xcprod_spec (@cprod_by gTH gTK H K gz isoZ).
|
Inductive
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprod_spec
| |
xcprodP: 'Z(H) \isog 'Z(K) -> xcprod_spec xcprod.
Proof. by rewrite /xcprod => isoZ; move: xcprod_subproof; rewrite isoZ. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
xcprodP
| |
isog_xcprod(rT : finGroupType) (GH GK G : {group rT}) :
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
GH \isog H -> GK \isog K -> GH \* GK = G -> 'Z(GH) = 'Z(GK) ->
G \isog [set: xcprod].
Proof.
move=> AutZinH isoGH isoGK defG eqZGHK; have [_ _ cGHK] := cprodP defG.
have [|gz isoZ] := xcprodP.
have [[fH injfH <-] [fK injfK <-]] := (isogP isoGH, isogP isoGK).
rewrite -!injm_center -?(isog_transl _ (sub_isog _ _)) ?center_sub //=.
by rewrite eqZGHK sub_isog ?center_sub.
rewrite (isog_cprod_by _ defG) //.
by apply/eqP; rewrite eqEsubset setIS // subsetI {2}eqZGHK !center_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
isog_xcprod
| |
ncprod_defn : finGroupType :=
if n is n'.+1 then xcprod G [set: ncprod_def n']
else subg_of 'Z(G).
Fact ncprod_key : unit. Proof. by []. Qed.
|
Fixpoint
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ncprod_def
| |
ncprod:= locked_with ncprod_key ncprod_def.
Local Notation G_ n := [set: gsort (ncprod n)].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ncprod
| |
ncprod0: G_ 0 \isog 'Z(G).
Proof. by rewrite [ncprod]unlock isog_sym isog_subg. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ncprod0
| |
center_ncprod0: 'Z(G_ 0) = G_ 0.
Proof. by apply: center_idP; rewrite (isog_abelian ncprod0) center_abelian. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_ncprod0
| |
center_ncprodn : 'Z(G_ n) \isog 'Z(G).
Proof.
elim: n => [|n]; first by rewrite center_ncprod0 ncprod0.
rewrite [ncprod]unlock=> /isog_symr/xcprodP[gz isoZ] /=.
by rewrite -cpairg1_center isog_sym sub_isog ?center_sub ?injm_cpairg1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
center_ncprod
| |
ncprodSn : xcprod_spec G [set: ncprod n] (ncprod n.+1).
Proof.
by have:= xcprodP (isog_symr (center_ncprod n)); rewrite [ncprod]unlock.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ncprodS
| |
ncprod1: G_ 1 \isog G.
Proof.
case: ncprodS => gz isoZ; rewrite isog_sym /= -im_cpair.
rewrite mulGSid /=; first by rewrite sub_isog ?injm_cpairg1.
rewrite -{3}center_ncprod0 injm_center ?injm_cpair1g //.
by rewrite -cpair_center_id center_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
ncprod1
| |
Aut_ncprod_fulln :
Aut_in (Aut G) 'Z(G) \isog Aut 'Z(G) ->
Aut_in (Aut (G_ n)) 'Z(G_ n) \isog Aut 'Z(G_ n).
Proof.
move=> AutZinG; elim: n => [|n IHn].
by rewrite center_ncprod0; apply/Aut_sub_fullP=> // g injg gG0; exists g.
by case: ncprodS => gz isoZ; apply: Aut_cprod_by_full.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] |
solvable/center.v
|
Aut_ncprod_full
| |
derived_atn (gT : finGroupType) (A : {set gT}) :=
iter n (fun B => [~: B, B]) A.
Arguments derived_at n%_N {gT} A%_g : simpl never.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
derived_at
| |
derg0A : A^`(0) = A. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
derg0
| |
derg1A : A^`(1) = [~: A, A]. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
derg1
| |
dergSnn A : A^`(n.+1) = [~: A^`(n), A^`(n)]. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
dergSn
| |
der_group_setG n : group_set G^`(n).
Proof. by case: n => [|n]; apply: groupP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_group_set
| |
derived_at_groupG n := Group (der_group_set G n).
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
derived_at_group
| |
conjg_mulRx y : x ^ y = x * [~ x, y].
Proof. by rewrite mulKVg. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
conjg_mulR
| |
conjg_Rmulx y : x ^ y = [~ y, x^-1] * x.
Proof. by rewrite commgEr invgK mulgKV. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
conjg_Rmul
| |
commMgJx y z : [~ x * y, z] = [~ x, z] ^ y * [~ y, z].
Proof. by rewrite !commgEr conjgM mulgA -conjMg mulgK. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commMgJ
| |
commgMJx y z : [~ x, y * z] = [~ x, z] * [~ x, y] ^ z.
Proof. by rewrite !commgEl conjgM -mulgA -conjMg mulKVg. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commgMJ
| |
commMgRx y z : [~ x * y, z] = [~ x, z] * [~ x, z, y] * [~ y, z].
Proof. by rewrite commMgJ conjg_mulR. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commMgR
| |
commgMRx y z : [~ x, y * z] = [~ x, z] * [~ x, y] * [~ x, y, z].
Proof. by rewrite commgMJ conjg_mulR mulgA. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commgMR
| |
Hall_Witt_identityx y z :
[~ x, y^-1, z] ^ y * [~ y, z^-1, x] ^ z * [~ z, x^-1, y] ^ x = 1.
Proof.
pose a x y z : gT := x * z * y ^ x.
suffices{x y z} hw_aux x y z: [~ x, y^-1, z] ^ y = (a x y z)^-1 * (a y z x).
by rewrite !hw_aux; move: a {hw_aux} => a; rewrite 2!mulgA !mulgK mulVg.
by rewrite commgEr conjMg -conjgM -conjg_Rmul conjgE !invMg !mulgA.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
Hall_Witt_identity
| |
commVg: [~ x^-1, y] = [~ x, y]^-1.
Proof.
apply/eqP; rewrite commgEl eq_sym eq_invg_mul invgK mulgA -cxz.
by rewrite -conjg_mulR -conjMg mulgV conj1g.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commVg
| |
commXg: [~ x ^+ i, y] = [~ x, y] ^+ i.
Proof.
elim: i => [|i' IHi]; first exact: comm1g.
by rewrite !expgS commMgJ /conjg commuteX // mulKg IHi.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commXg
| |
commgV: [~ x, y^-1] = [~ x, y]^-1.
Proof. by rewrite -invg_comm commVg -(invg_comm x y) ?invgK. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commgV
| |
commgX: [~ x, y ^+ i] = [~ x, y] ^+ i.
Proof. by rewrite -invg_comm commXg -(invg_comm x y) ?expgVn ?invgK. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commgX
| |
commXXg: [~ x ^+ i, y ^+ j] = [~ x, y] ^+ (i * j).
Proof. by rewrite expgM commgX commXg //; apply: commuteX. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commXXg
| |
expMg_Rmul: (y * x) ^+ i = y ^+ i * x ^+ i * [~ x, y] ^+ 'C(i, 2).
Proof.
rewrite -bin2_sum; symmetry.
elim: i => [|k IHk] /=; first by rewrite big_geq ?mulg1.
rewrite big_nat_recr //= addnC expgD !expgS -{}IHk !mulgA; congr (_ * _).
by rewrite -!mulgA commuteX2 // -commgX // [mul y]lock 3!mulgA -commgC.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
expMg_Rmul
| |
commG1A : [~: A, 1] = 1.
Proof. by apply/commG1P; rewrite centsC sub1G. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commG1
|
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