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 |
|---|---|---|---|---|---|---|
Frobenius_action:=
[/\ [faithful G, on S | to],
[transitive G, on S | to],
{in G^#, forall x, #|'Fix_(S | to)[x]| <= 1},
H != 1
& exists2 u, u \in S & H = 'C_G[u | to]].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_action
| |
has_Frobenius_actionG H : Prop :=
hasFrobeniusAction sT S to of @Frobenius_action G H sT S to.
|
Variant
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
has_Frobenius_action
| |
semiregular1lH : semiregular 1 H.
Proof. by move=> x _ /=; rewrite setI1g. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiregular1l
| |
semiregular1rK : semiregular K 1.
Proof. by move=> x; rewrite setDv inE. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiregular1r
| |
semiregular_symH K : semiregular K H -> semiregular H K.
Proof.
move=> regH x /setD1P[ntx Kx]; apply: contraNeq ntx.
rewrite -subG1 -setD_eq0 -setIDAC => /set0Pn[y /setIP[Hy cxy]].
by rewrite (sameP eqP set1gP) -(regH y Hy) inE Kx cent1C.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiregular_sym
| |
semiregularSK1 K2 A1 A2 :
K1 \subset K2 -> A1 \subset A2 -> semiregular K2 A2 -> semiregular K1 A1.
Proof.
move=> sK12 sA12 regKA2 x /setD1P[ntx /(subsetP sA12)A2x].
by apply/trivgP; rewrite -(regKA2 x) ?inE ?ntx ?setSI.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiregularS
| |
semiregular_primeH K : semiregular K H -> semiprime K H.
Proof.
move=> regH x Hx; apply/eqP; rewrite eqEsubset {1}regH // sub1G.
by rewrite -cent_set1 setIS ?centS // sub1set; case/setD1P: Hx.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiregular_prime
| |
semiprime_regularH K : semiprime K H -> 'C_K(H) = 1 -> semiregular K H.
Proof. by move=> prKH tiKcH x Hx; rewrite prKH. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiprime_regular
| |
semiprimeSK1 K2 A1 A2 :
K1 \subset K2 -> A1 \subset A2 -> semiprime K2 A2 -> semiprime K1 A1.
Proof.
move=> sK12 sA12 prKA2 x /setD1P[ntx A1x].
apply/eqP; rewrite eqEsubset andbC -{1}cent_set1 setIS ?centS ?sub1set //=.
rewrite -(setIidPl sK12) -!setIA prKA2 ?setIS ?centS //.
by rewrite !inE ntx (subsetP sA12).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiprimeS
| |
cent_semiprimeH K X :
semiprime K H -> X \subset H -> X :!=: 1 -> 'C_K(X) = 'C_K(H).
Proof.
move=> prKH sXH /trivgPn[x Xx ntx]; apply/eqP.
rewrite eqEsubset -{1}(prKH x) ?inE ?(subsetP sXH) ?ntx //=.
by rewrite -cent_cycle !setIS ?centS ?cycle_subG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
cent_semiprime
| |
stab_semiprimeH K X :
semiprime K H -> X \subset K -> 'C_H(X) != 1 -> 'C_H(X) = H.
Proof.
move=> prKH sXK ntCHX; apply/setIidPl; rewrite centsC -subsetIidl.
rewrite -{2}(setIidPl sXK) -setIA -(cent_semiprime prKH _ ntCHX) ?subsetIl //.
by rewrite !subsetI subxx sXK centsC subsetIr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
stab_semiprime
| |
cent_semiregularH K X :
semiregular K H -> X \subset H -> X :!=: 1 -> 'C_K(X) = 1.
Proof.
move=> regKH sXH /trivgPn[x Xx ntx]; apply/trivgP.
rewrite -(regKH x) ?inE ?(subsetP sXH) ?ntx ?setIS //=.
by rewrite -cent_cycle centS ?cycle_subG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
cent_semiregular
| |
regular_norm_dvd_predK H :
H \subset 'N(K) -> semiregular K H -> #|H| %| #|K|.-1.
Proof.
move=> nKH regH; have actsH: [acts H, on K^# | 'J] by rewrite astabsJ normD1.
rewrite (cardsD1 1 K) group1 -(acts_sum_card_orbit actsH) /=.
rewrite (eq_bigr (fun _ => #|H|)) ?sum_nat_const ?dvdn_mull //.
move=> _ /imsetP[x /setIdP[ntx Kx] ->]; rewrite card_orbit astab1J.
rewrite ['C_H[x]](trivgP _) ?indexg1 //=.
apply/subsetP=> y /setIP[Hy cxy]; apply: contraR ntx => nty.
by rewrite -[[set 1]](regH y) inE ?nty // Kx cent1C.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
regular_norm_dvd_pred
| |
regular_norm_coprimeK H :
H \subset 'N(K) -> semiregular K H -> coprime #|K| #|H|.
Proof.
move=> nKH regH.
by rewrite (coprime_dvdr (regular_norm_dvd_pred nKH regH)) ?coprimenP.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
regular_norm_coprime
| |
semiregularJK H x : semiregular K H -> semiregular (K :^ x) (H :^ x).
Proof.
move=> regH yx; rewrite -conjD1g => /imsetP[y Hy ->].
by rewrite cent1J -conjIg regH ?conjs1g.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiregularJ
| |
semiprimeJK H x : semiprime K H -> semiprime (K :^ x) (H :^ x).
Proof.
move=> prH yx; rewrite -conjD1g => /imsetP[y Hy ->].
by rewrite cent1J centJ -!conjIg prH.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiprimeJ
| |
normedTI_PA G L :
reflect [/\ A != set0, L \subset 'N_G(A)
& {in G, forall g, ~~ [disjoint A & A :^ g] -> g \in L}]
(normedTI A G L).
Proof.
apply: (iffP and3P) => [[nzA /trivIsetP tiAG /eqP <-] | [nzA sLN tiAG]].
split=> // g Gg; rewrite inE Gg (sameP normP eqP) /= eq_sym; apply: contraR.
by apply: tiAG; rewrite ?mem_orbit ?orbit_refl.
have [/set0Pn[a Aa] /subsetIP[_ nAL]] := (nzA, sLN); split=> //; last first.
rewrite eqEsubset sLN andbT; apply/subsetP=> x /setIP[Gx nAx].
by apply/tiAG/pred0Pn=> //; exists a; rewrite /= (normP nAx) Aa.
apply/trivIsetP=> _ _ /imsetP[x Gx ->] /imsetP[y Gy ->]; apply: contraR.
rewrite -setI_eq0 -(mulgKV x y) conjsgM; set g := (y * x^-1)%g.
have Gg: g \in G by rewrite groupMl ?groupV.
rewrite -conjIg (inj_eq (act_inj 'Js x)) (eq_sym A) (sameP eqP normP).
by rewrite -cards_eq0 cardJg cards_eq0 setI_eq0 => /tiAG/(subsetP nAL)->.
Qed.
Arguments normedTI_P {A G L}.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
normedTI_P
| |
normedTI_memJ_PA G L :
reflect [/\ A != set0, L \subset G
& {in A & G, forall a g, (a ^ g \in A) = (g \in L)}]
(normedTI A G L).
Proof.
apply: (iffP normedTI_P) => [[-> /subsetIP[sLG nAL] tiAG] | [-> sLG tiAG]].
split=> // a g Aa Gg; apply/idP/idP=> [Aag | Lg]; last first.
by rewrite memJ_norm ?(subsetP nAL).
by apply/tiAG/pred0Pn=> //; exists (a ^ g)%g; rewrite /= Aag memJ_conjg.
split=> // [ | g Gg /pred0Pn[ag /=]]; last first.
by rewrite andbC => /andP[/imsetP[a Aa ->]]; rewrite tiAG.
apply/subsetP=> g Lg; have Gg := subsetP sLG g Lg.
by rewrite !inE Gg; apply/subsetP=> _ /imsetP[a Aa ->]; rewrite tiAG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
normedTI_memJ_P
| |
partition_class_supportA G :
A != set0 -> trivIset (A :^: G) -> partition (A :^: G) (class_support A G).
Proof.
rewrite /partition cover_imset -class_supportEr eqxx => nzA ->.
by apply: contra nzA => /imsetP[x _ /eqP]; rewrite eq_sym -!cards_eq0 cardJg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
partition_class_support
| |
partition_normedTIA G L :
normedTI A G L -> partition (A :^: G) (class_support A G).
Proof. by case/and3P=> ntA tiAG _; apply: partition_class_support. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
partition_normedTI
| |
card_support_normedTIA G L :
normedTI A G L -> #|class_support A G| = (#|A| * #|G : L|)%N.
Proof.
case/and3P=> ntA tiAG /eqP <-; rewrite -card_conjugates mulnC.
apply: card_uniform_partition (partition_class_support ntA tiAG).
by move=> _ /imsetP[y _ ->]; rewrite cardJg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
card_support_normedTI
| |
normedTI_SA B G L :
A != set0 -> L \subset 'N(A) -> A \subset B -> normedTI B G L ->
normedTI A G L.
Proof.
move=> nzA /subsetP nAL /subsetP sAB /normedTI_memJ_P[nzB sLG tiB].
apply/normedTI_memJ_P; split=> // a x Aa Gx.
by apply/idP/idP => [Aax | /nAL/memJ_norm-> //]; rewrite -(tiB a) ?sAB.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
normedTI_S
| |
cent1_normedTIA G L :
normedTI A G L -> {in A, forall x, 'C_G[x] \subset L}.
Proof.
case/normedTI_memJ_P=> [_ _ tiAG] x Ax; apply/subsetP=> y /setIP[Gy cxy].
by rewrite -(tiAG x) // /(x ^ y) -(cent1P cxy) mulKg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
cent1_normedTI
| |
Frobenius_actionPG H :
reflect (has_Frobenius_action G H) [Frobenius G with complement H].
Proof.
apply: (iffP andP) => [[neqHG] | [sT S to [ffulG transG regG ntH [u Su defH]]]].
case/normedTI_P=> nzH /subsetIP[sHG _] tiHG.
suffices: Frobenius_action G H (rcosets H G) 'Rs by apply: hasFrobeniusAction.
pose Hfix x := 'Fix_(rcosets H G | 'Rs)[x].
have regG: {in G^#, forall x, #|Hfix x| <= 1}.
move=> x /setD1P[ntx Gx].
apply: wlog_neg; rewrite -ltnNge => /ltnW/card_gt0P/=[Hy].
rewrite -(cards1 Hy) => /setIP[/imsetP[y Gy ->{Hy}] cHyx].
apply/subset_leq_card/subsetP=> _ /setIP[/imsetP[z Gz ->] cHzx].
rewrite -!sub_astab1 !astab1_act !sub1set astab1Rs in cHyx cHzx *.
rewrite !rcosetE; apply/set1P/rcoset_eqP; rewrite mem_rcoset.
apply: tiHG; [by rewrite !in_group | apply/pred0Pn; exists (x ^ y^-1)].
by rewrite conjD1g !inE conjg_eq1 ntx -mem_conjg cHyx conjsgM memJ_conjg.
have ntH: H :!=: 1 by rewrite -subG1 -setD_eq0.
split=> //; first 1 last; first exact: transRs_rcosets.
by exists (val H); rewrite ?orbit_refl // astab1Rs (setIidPr sHG).
apply/subsetP=> y /setIP[Gy cHy]; apply: contraR neqHG => nt_y.
rewrite (index1g sHG) //; apply/eqP; rewrite eqn_leq indexg_gt0 andbT.
apply: leq_trans (regG y _); last by rewrite setDE 2!inE Gy nt_y /=.
by rewrite /Hfix (setIidPl _) -1?astabC ?sub1set.
have sHG: H \subset G by rewrite defH subsetIl.
split.
apply: contraNneq ntH => /= defG.
suffices defS: S = [set u] by rewrite -(trivgP ffulG) /= defS d
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_actionP
| |
FrobeniusWker: [Frobenius G with kernel K].
Proof. by apply/existsP; exists H. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
FrobeniusWker
| |
FrobeniusWcompl: [Frobenius G with complement H].
Proof. by case/andP: frobG. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
FrobeniusWcompl
| |
FrobeniusW: [Frobenius G].
Proof. by apply/existsP; exists H; apply: FrobeniusWcompl. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
FrobeniusW
| |
Frobenius_context:
[/\ K ><| H = G, K :!=: 1, H :!=: 1, K \proper G & H \proper G].
Proof.
have [/eqP defG neqHG ntH _] := and4P frobG; rewrite setD_eq0 subG1 in ntH.
have ntK: K :!=: 1 by apply: contraNneq neqHG => K1; rewrite -defG K1 sdprod1g.
rewrite properEcard properEneq neqHG; have /mulG_sub[-> ->] := sdprodW defG.
by rewrite -(sdprod_card defG) ltn_Pmulr ?cardG_gt1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_context
| |
Frobenius_partition: partition (gval K |: (H^# :^: K)) G.
Proof.
have [/eqP defG _ tiHG] := and3P frobG; have [_ tiH1G /eqP defN] := and3P tiHG.
have [[_ /mulG_sub[sKG sHG] nKH tiKH] mulHK] := (sdprodP defG, sdprodWC defG).
set HG := H^# :^: K; set KHG := _ |: _.
have defHG: HG = H^# :^: G.
have: 'C_G[H^# | 'Js] * K = G by rewrite astab1Js defN mulHK.
move/subgroup_transitiveP/atransP.
by apply; rewrite ?atrans_orbit ?orbit_refl.
have /and3P[defHK _ nzHG] := partition_normedTI tiHG.
rewrite -defHG in defHK nzHG tiH1G.
have [tiKHG HG'K]: trivIset KHG /\ gval K \notin HG.
apply: trivIsetU1 => // _ /imsetP[x Kx ->]; rewrite -setI_eq0.
by rewrite -(conjGid Kx) -conjIg setIDA tiKH setDv conj0g.
rewrite /partition andbC tiKHG !inE negb_or nzHG eq_sym -card_gt0 cardG_gt0 /=.
rewrite eqEcard; apply/andP; split.
rewrite /cover big_setU1 //= subUset sKG -/(cover HG) (eqP defHK).
by rewrite class_support_subG // (subset_trans _ sHG) ?subD1set.
rewrite -(eqnP tiKHG) big_setU1 //= (eqnP tiH1G) (eqP defHK).
rewrite (card_support_normedTI tiHG) -(Lagrange sHG) (cardsD1 1) group1 mulSn.
by rewrite leq_add2r -mulHK indexMg -indexgI tiKH indexg1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_partition
| |
Frobenius_cent1_ker: {in K^#, forall x, 'C_G[x] \subset K}.
Proof.
have [/eqP defG _ /normedTI_memJ_P[_ _ tiHG]] := and3P frobG.
move=> x /setD1P[ntx Kx]; have [_ /mulG_sub[sKG _] _ tiKH] := sdprodP defG.
have [/eqP <- _ _] := and3P Frobenius_partition; rewrite big_distrl /=.
apply/bigcupsP=> _ /setU1P[|/imsetP[y Ky]] ->; first exact: subsetIl.
apply: contraR ntx => /subsetPn[z]; rewrite inE mem_conjg => /andP[Hzy cxz] _.
rewrite -(conjg_eq1 x y^-1) -in_set1 -set1gE -tiKH inE andbC.
rewrite -(tiHG _ _ Hzy) ?(subsetP sKG) ?in_group // Ky andbT -conjJg.
by rewrite /(z ^ x) (cent1P cxz) mulKg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_cent1_ker
| |
Frobenius_reg_ker: semiregular K H.
Proof.
move=> x /setD1P[ntx Hx].
apply/trivgP/subsetP=> y /setIP[Ky cxy]; apply: contraR ntx => nty.
have K1y: y \in K^# by rewrite inE nty.
have [/eqP/sdprod_context[_ sHG _ _ tiKH] _] := andP frobG.
suffices: x \in K :&: H by rewrite tiKH inE.
by rewrite inE (subsetP (Frobenius_cent1_ker K1y)) // inE cent1C (subsetP sHG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_reg_ker
| |
Frobenius_reg_compl: semiregular H K.
Proof. by apply: semiregular_sym; apply: Frobenius_reg_ker. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_reg_compl
| |
Frobenius_dvd_ker1: #|H| %| #|K|.-1.
Proof.
apply: regular_norm_dvd_pred Frobenius_reg_ker.
by have[/sdprodP[]] := Frobenius_context.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_dvd_ker1
| |
ltn_odd_Frobenius_ker: odd #|G| -> #|H|.*2 < #|K|.
Proof.
move/oddSg=> oddG.
have [/sdprodW/mulG_sub[sKG sHG] ntK _ _ _] := Frobenius_context.
by rewrite dvdn_double_ltn ?oddG ?cardG_gt1 ?Frobenius_dvd_ker1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
ltn_odd_Frobenius_ker
| |
Frobenius_index_dvd_ker1: #|G : K| %| #|K|.-1.
Proof.
have[defG _ _ /andP[sKG _] _] := Frobenius_context.
by rewrite -divgS // -(sdprod_card defG) mulKn ?Frobenius_dvd_ker1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_index_dvd_ker1
| |
Frobenius_coprime: coprime #|K| #|H|.
Proof. by rewrite (coprime_dvdr Frobenius_dvd_ker1) ?coprimenP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_coprime
| |
Frobenius_trivg_cent: 'C_K(H) = 1.
Proof.
by apply: (cent_semiregular Frobenius_reg_ker); case: Frobenius_context.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_trivg_cent
| |
Frobenius_index_coprime: coprime #|K| #|G : K|.
Proof. by rewrite (coprime_dvdr Frobenius_index_dvd_ker1) ?coprimenP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_index_coprime
| |
Frobenius_ker_Hall: Hall G K.
Proof.
have [_ _ _ /andP[sKG _] _] := Frobenius_context.
by rewrite /Hall sKG Frobenius_index_coprime.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_ker_Hall
| |
Frobenius_compl_Hall: Hall G H.
Proof.
have [defG _ _ _ _] := Frobenius_context.
by rewrite -(sdprod_Hall defG) Frobenius_ker_Hall.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_compl_Hall
| |
normedTI_Jx A G L : normedTI (A :^ x) (G :^ x) (L :^ x) = normedTI A G L.
Proof.
rewrite {1}/normedTI normJ -conjIg -(conj0g x) !(can_eq (conjsgK x)).
congr [&& _, _ == _ & _]; rewrite /cover (reindex_inj (@conjsg_inj _ x)).
by apply: eq_big => Hy; rewrite ?orbit_conjsg ?cardJg.
by rewrite bigcupJ cardJg (eq_bigl _ _ (orbit_conjsg _ _ _ _)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
normedTI_J
| |
FrobeniusJcomplx G H :
[Frobenius G :^ x with complement H :^ x] = [Frobenius G with complement H].
Proof.
by congr (_ && _); rewrite ?(can_eq (conjsgK x)) // -conjD1g normedTI_J.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
FrobeniusJcompl
| |
FrobeniusJx G K H :
[Frobenius G :^ x = K :^ x ><| H :^ x] = [Frobenius G = K ><| H].
Proof.
by congr (_ && _); rewrite ?FrobeniusJcompl // -sdprodJ (can_eq (conjsgK x)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
FrobeniusJ
| |
FrobeniusJkerx G K :
[Frobenius G :^ x with kernel K :^ x] = [Frobenius G with kernel K].
Proof.
apply/existsP/existsP=> [] [H]; last by exists (H :^ x)%G; rewrite FrobeniusJ.
by rewrite -(conjsgKV x H) FrobeniusJ; exists (H :^ x^-1)%G.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
FrobeniusJker
| |
FrobeniusJgroupx G : [Frobenius G :^ x] = [Frobenius G].
Proof.
apply/existsP/existsP=> [] [H].
by rewrite -(conjsgKV x H) FrobeniusJcompl; exists (H :^ x^-1)%G.
by exists (H :^ x)%G; rewrite FrobeniusJcompl.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
FrobeniusJgroup
| |
Frobenius_ker_dvd_ker1G K :
[Frobenius G with kernel K] -> #|G : K| %| #|K|.-1.
Proof. by case/existsP=> H; apply: Frobenius_index_dvd_ker1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_ker_dvd_ker1
| |
Frobenius_ker_coprimeG K :
[Frobenius G with kernel K] -> coprime #|K| #|G : K|.
Proof. by case/existsP=> H; apply: Frobenius_index_coprime. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_ker_coprime
| |
Frobenius_semiregularPG K H :
K ><| H = G -> K :!=: 1 -> H :!=: 1 ->
reflect (semiregular K H) [Frobenius G = K ><| H].
Proof.
move=> defG ntK ntH.
apply: (iffP idP) => [|regG]; first exact: Frobenius_reg_ker.
have [nsKG sHG defKH nKH tiKH]:= sdprod_context defG; have [sKG _]:= andP nsKG.
apply/and3P; split; first by rewrite defG.
by rewrite eqEcard sHG -(sdprod_card defG) -ltnNge ltn_Pmull ?cardG_gt1.
apply/normedTI_memJ_P; rewrite setD_eq0 subG1 sHG -defKH -(normC nKH).
split=> // z _ /setD1P[ntz Hz] /mulsgP[y x Hy Kx ->]; rewrite groupMl // !inE.
rewrite conjg_eq1 ntz; apply/idP/idP=> [Hzxy | Hx]; last by rewrite !in_group.
apply: (subsetP (sub1G H)); have Hzy: z ^ y \in H by apply: groupJ.
rewrite -(regG (z ^ y)); last by apply/setD1P; rewrite conjg_eq1.
rewrite inE Kx cent1C (sameP cent1P commgP) -in_set1 -[[set 1]]tiKH inE /=.
rewrite andbC groupM ?groupV -?conjgM //= commgEr groupMr //.
by rewrite memJ_norm ?(subsetP nKH) ?groupV.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_semiregularP
| |
prime_FrobeniusPG K H :
K :!=: 1 -> prime #|H| ->
reflect (K ><| H = G /\ 'C_K(H) = 1) [Frobenius G = K ><| H].
Proof.
move=> ntK H_pr; have ntH: H :!=: 1 by rewrite -cardG_gt1 prime_gt1.
have [defG | not_sdG] := eqVneq (K ><| H) G; last first.
by apply: (iffP andP) => [] [defG]; rewrite defG ?eqxx in not_sdG.
apply: (iffP (Frobenius_semiregularP defG ntK ntH)) => [regH | [_ regH x]].
split=> //; have [x defH] := cyclicP (prime_cyclic H_pr).
by rewrite defH cent_cycle regH // !inE defH cycle_id andbT -cycle_eq1 -defH.
case/setD1P=> nt_x Hx; apply/trivgP; rewrite -regH setIS //= -cent_cycle.
by rewrite centS // prime_meetG // (setIidPr _) ?cycle_eq1 ?cycle_subG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
prime_FrobeniusP
| |
Frobenius_sublG K K1 H :
K1 :!=: 1 -> K1 \subset K -> H \subset 'N(K1) -> [Frobenius G = K ><| H] ->
[Frobenius K1 <*> H = K1 ><| H].
Proof.
move=> ntK1 sK1K nK1H frobG; have [_ _ ntH _ _] := Frobenius_context frobG.
apply/Frobenius_semiregularP=> //.
by rewrite sdprodEY ?coprime_TIg ?(coprimeSg sK1K) ?(Frobenius_coprime frobG).
by move=> x /(Frobenius_reg_ker frobG) cKx1; apply/trivgP; rewrite -cKx1 setSI.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_subl
| |
Frobenius_subrG K H H1 :
H1 :!=: 1 -> H1 \subset H -> [Frobenius G = K ><| H] ->
[Frobenius K <*> H1 = K ><| H1].
Proof.
move=> ntH1 sH1H frobG; have [defG ntK _ _ _] := Frobenius_context frobG.
apply/Frobenius_semiregularP=> //.
have [_ _ /(subset_trans sH1H) nH1K tiHK] := sdprodP defG.
by rewrite sdprodEY //; apply/trivgP; rewrite -tiHK setIS.
by apply: sub_in1 (Frobenius_reg_ker frobG); apply/subsetP/setSD.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_subr
| |
Frobenius_kerPG K :
reflect [/\ K :!=: 1, K \proper G, K <| G
& {in K^#, forall x, 'C_G[x] \subset K}]
[Frobenius G with kernel K].
Proof.
apply: (iffP existsP) => [[H frobG] | [ntK ltKG nsKG regK]].
have [/sdprod_context[nsKG _ _ _ _] ntK _ ltKG _] := Frobenius_context frobG.
by split=> //; apply: Frobenius_cent1_ker frobG.
have /andP[sKG nKG] := nsKG.
have hallK: Hall G K.
rewrite /Hall sKG //= coprime_sym coprime_pi' //.
apply: sub_pgroup (pgroup_pi K) => p; have [P sylP] := Sylow_exists p G.
have [[sPG pP p'GiP] sylPK] := (and3P sylP, Hall_setI_normal nsKG sylP).
rewrite -p_rank_gt0 -(rank_Sylow sylPK) rank_gt0 => ntPK.
rewrite inE /= -p'natEpi // (pnat_dvd _ p'GiP) ?indexgS //.
have /trivgPn[z]: P :&: K :&: 'Z(P) != 1.
by rewrite meet_center_nil ?(pgroup_nil pP) ?(normalGI sPG nsKG).
rewrite !inE -andbA -sub_cent1=> /and4P[_ Kz _ cPz] ntz.
by apply: subset_trans (regK z _); [apply/subsetIP | apply/setD1P].
have /splitsP[H /complP[tiKH defG]] := SchurZassenhaus_split hallK nsKG.
have [_ sHG] := mulG_sub defG; have nKH := subset_trans sHG nKG.
exists H; apply/Frobenius_semiregularP; rewrite ?sdprodE //.
by apply: contraNneq (proper_subn ltKG) => H1; rewrite -defG H1 mulg1.
apply: semiregular_sym => x Kx; apply/trivgP; rewrite -tiKH.
by rewrite subsetI subsetIl (subset_trans _ (regK x _)) ?setSI.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_kerP
| |
set_Frobenius_complG K H :
K ><| H = G -> [Frobenius G with kernel K] -> [Frobenius G = K ><| H].
Proof.
move=> defG /Frobenius_kerP[ntK ltKG _ regKG].
apply/Frobenius_semiregularP=> //.
by apply: contraTneq ltKG => H_1; rewrite -defG H_1 sdprodg1 properxx.
apply: semiregular_sym => y /regKG sCyK.
have [_ sHG _ _ tiKH] := sdprod_context defG.
by apply/trivgP; rewrite /= -(setIidPr sHG) setIAC -tiKH setSI.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
set_Frobenius_compl
| |
Frobenius_kerSG K G1 :
G1 \subset G -> K \proper G1 ->
[Frobenius G with kernel K] -> [Frobenius G1 with kernel K].
Proof.
move=> sG1G ltKG1 /Frobenius_kerP[ntK _ /andP[_ nKG] regKG].
apply/Frobenius_kerP; rewrite /normal proper_sub // (subset_trans sG1G) //.
by split=> // x /regKG; apply: subset_trans; rewrite setSI.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_kerS
| |
Frobenius_action_kernel_defG H K sT S to :
K ><| H = G -> @Frobenius_action _ G H sT S to ->
K :=: 1 :|: [set x in G | 'Fix_(S | to)[x] == set0].
Proof.
move=> defG FrobG.
have partG: partition (gval K |: (H^# :^: K)) G.
apply: Frobenius_partition; apply/andP; rewrite defG; split=> //.
by apply/Frobenius_actionP; apply: hasFrobeniusAction FrobG.
have{FrobG} [ffulG transG regG ntH [u Su defH]]:= FrobG.
apply/setP=> x /[!inE]; have [-> | ntx] := eqVneq; first exact: group1.
rewrite /= -(cover_partition partG) /cover.
have neKHy y: gval K <> H^# :^ y.
by move/setP/(_ 1); rewrite group1 conjD1g setD11.
rewrite big_setU1 /= ?inE; last by apply/imsetP=> [[y _ /neKHy]].
have [nsKG sHG _ _ tiKH] := sdprod_context defG; have [sKG nKG]:= andP nsKG.
symmetry; case Kx: (x \in K) => /=.
apply/set0Pn=> [[v /setIP[Sv]]]; have [y Gy ->] := atransP2 transG Su Sv.
rewrite -sub1set -astabC sub1set astab1_act mem_conjg => Hxy.
case/negP: ntx; rewrite -in_set1 -(conjgKV y x) -mem_conjgV conjs1g -tiKH.
by rewrite defH setIA inE -mem_conjg (setIidPl sKG) (normsP nKG) ?Kx.
apply/andP=> [[/bigcupP[_ /imsetP[y Ky ->] Hyx] /set0Pn[]]]; exists (to u y).
rewrite inE (actsP (atrans_acts transG)) ?(subsetP sKG) // Su.
rewrite -sub1set -astabC sub1set astab1_act.
by rewrite conjD1g defH conjIg !inE in Hyx; case/and3P: Hyx.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_action_kernel_def
| |
Frobenius_coprime_quotient(gT : finGroupType) (G K H N : {group gT}) :
K ><| H = G -> N <| G -> coprime #|K| #|H| /\ H :!=: 1%g ->
N \proper K /\ {in H^#, forall x, 'C_K[x] \subset N} ->
[Frobenius G / N = (K / N) ><| (H / N)]%g.
Proof.
move=> defG nsNG [coKH ntH] [ltNK regH].
have [[sNK _] [_ /mulG_sub[sKG sHG] _ _]] := (andP ltNK, sdprodP defG).
have [_ nNG] := andP nsNG; have nNH := subset_trans sHG nNG.
apply/Frobenius_semiregularP; first exact: quotient_coprime_sdprod.
- by rewrite quotient_neq1 ?(normalS _ sKG).
- by rewrite -(isog_eq1 (quotient_isog _ _)) ?coprime_TIg ?(coprimeSg sNK).
move=> _ /(subsetP (quotientD1 _ _))/morphimP[x nNx H1x ->].
rewrite -cent_cycle -quotient_cycle //=.
rewrite -strongest_coprime_quotient_cent ?cycle_subG //.
- by rewrite cent_cycle quotientS1 ?regH.
- by rewrite subIset ?sNK.
- rewrite (coprimeSg (subsetIl N _)) ?(coprimeSg sNK) ?(coprimegS _ coKH) //.
by rewrite cycle_subG; case/setD1P: H1x.
by rewrite orbC abelian_sol ?cycle_abelian.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_coprime_quotient
| |
injm_Frobenius_complH sGD injf :
[Frobenius G with complement H] -> [Frobenius f @* G with complement f @* H].
Proof.
case/andP=> neqGH /normedTI_P[nzH /subsetIP[sHG _] tiHG].
have sHD := subset_trans sHG sGD; have sH1D := subset_trans (subD1set H 1) sHD.
apply/andP; rewrite (can_in_eq (injmK injf)) //; split=> //.
apply/normedTI_P; rewrite normD1 -injmD1 // -!cards_eq0 card_injm // in nzH *.
rewrite subsetI normG morphimS //; split=> // _ /morphimP[x Dx Gx ->] ti'fHx.
rewrite mem_morphim ?tiHG //; apply: contra ti'fHx; rewrite -!setI_eq0 => tiHx.
by rewrite -morphimJ // -injmI ?conj_subG // (eqP tiHx) morphim0.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
injm_Frobenius_compl
| |
injm_FrobeniusH K sGD injf :
[Frobenius G = K ><| H] -> [Frobenius f @* G = f @* K ><| f @* H].
Proof.
case/andP=> /eqP defG frobG.
by apply/andP; rewrite (injm_sdprod _ injf defG) // eqxx injm_Frobenius_compl.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
injm_Frobenius
| |
injm_Frobenius_kerK sGD injf :
[Frobenius G with kernel K] -> [Frobenius f @* G with kernel f @* K].
Proof.
case/existsP=> H frobG; apply/existsP.
by exists (f @* H)%G; apply: injm_Frobenius.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
injm_Frobenius_ker
| |
injm_Frobenius_groupsGD injf : [Frobenius G] -> [Frobenius f @* G].
Proof.
case/existsP=> H frobG; apply/existsP; exists (f @* H)%G.
exact: injm_Frobenius_compl.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
injm_Frobenius_group
| |
Frobenius_Ldiv(gT : finGroupType) (G : {group gT}) n :
n %| #|G| -> n %| #|'Ldiv_n(G)|.
Proof.
move=> nG; move: {2}_.+1 (ltnSn (#|G| %/ n)) => mq.
elim: mq => // mq IHm in gT G n nG *; case/dvdnP: nG => q oG.
have [q_gt0 n_gt0] : 0 < q /\ 0 < n by apply/andP; rewrite -muln_gt0 -oG.
rewrite ltnS oG mulnK // => leqm.
have:= q_gt0; rewrite leq_eqVlt => /predU1P[q1 | lt1q].
rewrite -(mul1n n) q1 -oG (setIidPl _) //.
by apply/subsetP=> x Gx; rewrite inE -order_dvdn order_dvdG.
pose p := pdiv q; have pr_p: prime p by apply: pdiv_prime.
have lt1p: 1 < p := prime_gt1 pr_p; have p_gt0 := ltnW lt1p.
have{leqm} lt_qp_mq: q %/ p < mq by apply: leq_trans leqm; rewrite ltn_Pdiv.
have: n %| #|'Ldiv_(p * n)(G)|.
have: p * n %| #|G| by rewrite oG dvdn_pmul2r ?pdiv_dvd.
move/IHm=> IH; apply: dvdn_trans (IH _); first exact: dvdn_mull.
by rewrite oG divnMr.
rewrite -(cardsID 'Ldiv_n()) dvdn_addl.
rewrite -setIA ['Ldiv_n(_)](setIidPr _) //.
by apply/subsetP=> x; rewrite !inE -!order_dvdn; apply: dvdn_mull.
rewrite -setIDA; set A := _ :\: _.
have pA x: x \in A -> (#[x]`_p = n`_p * p)%N.
rewrite !inE -!order_dvdn => /andP[xn xnp].
rewrite !p_part // -expnSr; congr (p ^ _)%N; apply/eqP.
rewrite eqn_leq -{1}addn1 -(pfactorK 1 pr_p) -lognM ?expn1 // mulnC.
rewrite dvdn_leq_log ?muln_gt0 ?p_gt0 //= ltnNge; apply: contra xn => xn.
move: xnp; rewrite -[#[x]](partnC p) //.
rewrite !Gauss_dvd ?coprime_partC //; case/andP=> _.
rewrite p_part ?pfactor_dvdn // xn Gauss_dvdr // coprime_sym.
e
...
|
Theorem
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_Ldiv
| |
object_map:= forall gT : finGroupType, {set gT} -> {set gT}.
Bind Scope gFun_scope with object_map.
|
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
|
object_map
| |
group_valued:= forall gT (G : {group gT}), group_set (F G).
|
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
|
group_valued
| |
closed:= forall gT (G : {group gT}), F G \subset G.
|
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
|
closed
| |
continuous:=
forall gT hT (G : {group gT}) (phi : {morphism G >-> hT}),
phi @* F G \subset F (phi @* G).
|
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
|
continuous
| |
iso_continuous:=
forall gT hT (G : {group gT}) (phi : {morphism G >-> hT}),
'injm phi -> phi @* F G \subset F (phi @* G).
|
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
|
iso_continuous
| |
continuous_is_iso_continuous: continuous -> iso_continuous.
Proof. by move=> Fcont gT hT G phi inj_phi; apply: Fcont. 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
|
continuous_is_iso_continuous
| |
pcontinuous:=
forall gT hT (G D : {group gT}) (phi : {morphism D >-> hT}),
phi @* F G \subset F (phi @* G).
|
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
|
pcontinuous
| |
pcontinuous_is_continuous: pcontinuous -> continuous.
Proof. by move=> Fcont gT hT G; apply: Fcont. 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
|
pcontinuous_is_continuous
| |
hereditary:=
forall gT (H G : {group gT}), H \subset G -> F G :&: H \subset F H.
|
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
|
hereditary
| |
pcontinuous_is_hereditary: pcontinuous -> hereditary.
Proof.
move=> Fcont gT H G sHG; rewrite -{2}(setIidPl sHG) setIC.
by do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom ?Fcont.
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
|
pcontinuous_is_hereditary
| |
monotonic:=
forall gT (H G : {group gT}), H \subset G -> F H \subset F G.
|
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
|
monotonic
| |
comp: object_map := fun gT A => F1 (F2 A).
|
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
|
comp
| |
modulo: object_map :=
fun gT A => coset (F2 A) @*^-1 (F1 (A / (F2 A))).
|
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
|
modulo
| |
iso_map:= IsoMap {
apply : object_map;
_ : group_valued apply;
_ : closed apply;
_ : iso_continuous apply
}.
Local Coercion apply : iso_map >-> object_map.
|
Structure
|
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
|
iso_map
| |
map:= Map { iso_of_map : iso_map; _ : continuous iso_of_map }.
Local Coercion iso_of_map : map >-> iso_map.
|
Structure
|
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
|
map
| |
pmap:= Pmap { map_of_pmap : map; _ : hereditary map_of_pmap }.
Local Coercion map_of_pmap : pmap >-> map.
|
Structure
|
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
|
pmap
| |
mono_map:= MonoMap { map_of_mono : map; _ : monotonic map_of_mono }.
Local Coercion map_of_mono : mono_map >-> map.
|
Structure
|
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
|
mono_map
| |
pack_isoF Fcont Fgrp Fsub := @IsoMap F Fgrp Fsub Fcont.
|
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
|
pack_iso
| |
clone_iso(F : object_map) :=
fun Fgrp Fsub Fcont (isoF := @IsoMap F Fgrp Fsub Fcont) =>
fun isoF0 & phant_id (apply isoF0) F & phant_id isoF isoF0 => isoF.
|
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
|
clone_iso
| |
clone(F : object_map) :=
fun isoF & phant_id (apply isoF) F =>
fun (funF0 : map) & phant_id (apply funF0) F =>
fun Fcont (funF := @Map isoF Fcont) & phant_id funF0 funF => funF.
|
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
|
clone
| |
clone_pmap(F : object_map) :=
fun (funF : map) & phant_id (apply funF) F =>
fun (pfunF0 : pmap) & phant_id (apply pfunF0) F =>
fun Fher (pfunF := @Pmap funF Fher) & phant_id pfunF0 pfunF => pfunF.
|
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
|
clone_pmap
| |
clone_mono(F : object_map) :=
fun (funF : map) & phant_id (apply funF) F =>
fun (mfunF0 : mono_map) & phant_id (apply mfunF0) F =>
fun Fmon (mfunF := @MonoMap funF Fmon) & phant_id mfunF0 mfunF => mfunF.
|
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
|
clone_mono
| |
apply: iso_map >-> object_map.
|
Coercion
|
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
|
apply
| |
iso_of_map: map >-> iso_map.
|
Coercion
|
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
|
iso_of_map
| |
map_of_pmap: pmap >-> map.
|
Coercion
|
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
|
map_of_pmap
| |
map_of_mono: mono_map >-> map.
|
Coercion
|
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
|
map_of_mono
| |
continuous_is_iso_continuous: continuous >-> iso_continuous.
|
Coercion
|
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
|
continuous_is_iso_continuous
| |
pcontinuous_is_continuous: pcontinuous >-> continuous.
|
Coercion
|
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
|
pcontinuous_is_continuous
| |
pcontinuous_is_hereditary: pcontinuous >-> hereditary.
|
Coercion
|
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
|
pcontinuous_is_hereditary
| |
gFgroupset: group_set (F gT G). 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
|
gFgroupset
| |
gFgroup:= Group gFgroupset.
|
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
|
gFgroup
| |
gFmod_group(F1 : GFunctor.iso_map) (F2 : GFunctor.object_map)
(gT : finGroupType) (G : {group gT}) :=
[group of (F1 %% F2)%gF gT G].
|
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_group
| |
gFsubgT (G : {group gT}) : F gT G \subset G.
Proof. by case: F gT 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
|
gFsub
| |
gFsub_transgT (G : {group gT}) (A : {pred gT}) :
G \subset A -> F gT G \subset A.
Proof. exact/subset_trans/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
|
gFsub_trans
| |
gF1gT : F gT 1 = 1. Proof. exact/trivgP/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
|
gF1
| |
gFiso_cont: GFunctor.iso_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
|
gFiso_cont
| |
gFchargT (G : {group gT}) : F gT G \char G.
Proof.
apply/andP; split => //; first by apply: gFsub.
apply/forall_inP=> f Af; rewrite -{2}(im_autm Af) -(autmE Af).
by rewrite -morphimEsub ?gFsub ?gFiso_cont ?injm_autm.
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
| |
gFnormgT (G : {group gT}) : G \subset 'N(F gT G).
Proof. exact/char_norm/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
|
gFnorm
| |
gFnormsgT (G : {group gT}) : 'N(G) \subset 'N(F gT G).
Proof. exact/char_norms/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
|
gFnorms
|
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