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 /setId... | 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... | 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_... | 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... | 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 ... | 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 ... | 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]] ... | 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_cen... | 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 _ ... | 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; firs... | 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.
... | 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) ?(Frobeni... | 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 /... | 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=> //;... | 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 _ _ t... | 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 s... | 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_acti... | 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 s... | 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=> //.
a... | 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.
h... | 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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