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 |
|---|---|---|---|---|---|---|
norm_conj_centA G x : x \in 'C(A) ->
(A \subset 'N(G :^ x)) = (A \subset 'N(G)).
Proof. by move=> cAx; rewrite norm_conj_norm ?(subsetP (cent_sub A)). Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | norm_conj_cent | |
strongest_coprime_quotient_centA G H :
let R := H :&: [~: G, A] in
A \subset 'N(H) -> R \subset G -> coprime #|R| #|A| ->
solvable R || solvable A ->
'C_G(A) / H = 'C_(G / H)(A / H).
Proof.
move=> R nHA sRG coRA solRA.
have nRA: A \subset 'N(R) by rewrite normsI ?commg_normr.
apply/eqP; rewrite eqEs... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | strongest_coprime_quotient_cent | |
coprime_norm_quotient_centA G H :
A \subset 'N(G) -> A \subset 'N(H) -> coprime #|H| #|A| -> solvable H ->
'C_G(A) / H = 'C_(G / H)(A / H).
Proof.
move=> nGA nHA coHA solH; have sRH := subsetIl H [~: G, A].
rewrite strongest_coprime_quotient_cent ?(coprimeSg sRH) 1?(solvableS sRH) //.
by rewrite subIset // commg_... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | coprime_norm_quotient_cent | |
coprime_cent_mulGA G H :
A \subset 'N(G) -> A \subset 'N(H) -> G \subset 'N(H) ->
coprime #|H| #|A| -> solvable H ->
'C_(H * G)(A) = 'C_H(A) * 'C_G(A).
Proof.
move=> nHA nGA nHG coHA solH; rewrite -norm_joinEr //.
have nsHG: H <| H <*> G by rewrite /normal joing_subl join_subG normG.
rewrite -{2}(setIidPr (... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | coprime_cent_mulG | |
quotient_TI_subcentK G H :
G \subset 'N(K) -> G \subset 'N(H) -> K :&: H = 1 ->
'C_K(G) / H = 'C_(K / H)(G / H).
Proof.
move=> nGK nGH tiKH.
have tiHR: H :&: [~: K, G] = 1.
by apply/trivgP; rewrite /= setIC -tiKH setSI ?commg_subl.
apply: strongest_coprime_quotient_cent; rewrite ?tiHR ?sub1G ?solvable1 //.
by r... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | quotient_TI_subcent | |
external_action_im_coprime: coprime #|G'| #|A'|.
Proof. by rewrite !card_injm. Qed.
Let coGA' := external_action_im_coprime.
Let solG' : solvable G' := morphim_sol _ solG.
Let nGA' := im_sdpair_norm to. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | external_action_im_coprime | |
ext_coprime_Hall_exists:
exists2 H : {group gT}, pi.-Hall(G) H & [acts A, on H | to].
Proof.
have [H' hallH' nHA'] := coprime_Hall_exists pi nGA' coGA' solG'.
have sHG' := pHall_sub hallH'.
exists (inG @*^-1 H')%G => /=.
by rewrite -(morphim_invmE injG) -{1}(im_invm injG) morphim_pHall.
by rewrite actsEsd ?morphpre... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | ext_coprime_Hall_exists | |
ext_coprime_Hall_trans(H1 H2 : {group gT}) :
pi.-Hall(G) H1 -> [acts A, on H1 | to] ->
pi.-Hall(G) H2 -> [acts A, on H2 | to] ->
exists2 x, x \in 'C_(G | to)(A) & H1 :=: H2 :^ x.
Proof.
move=> hallH1 nH1A hallH2 nH2A.
have sH1G := pHall_sub hallH1; have sH2G := pHall_sub hallH2.
rewrite !actsEsd // in nH1A nH... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | ext_coprime_Hall_trans | |
ext_norm_conj_cent(H : {group gT}) x :
H \subset G -> x \in 'C_(G | to)(A) ->
[acts A, on H :^ x | to] = [acts A, on H | to].
Proof.
move=> sHG /setIP[Gx].
rewrite gacentEsd !actsEsd ?conj_subG ?morphimJ // 2!inE Gx /=.
exact: norm_conj_cent.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | ext_norm_conj_cent | |
ext_coprime_Hall_subset(X : {group gT}) :
X \subset G -> pi.-group X -> [acts A, on X | to] ->
exists H : {group gT},
[/\ pi.-Hall(G) H, [acts A, on H | to] & X \subset H].
Proof.
move=> sXG piX; rewrite actsEsd // => nXA'.
case: (coprime_Hall_subset nGA' coGA' solG' _ (morphim_pgroup _ piX) nXA').
exact: m... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | ext_coprime_Hall_subset | |
ext_coprime_quotient_cent(H : {group gT}) :
H \subset G -> [acts A, on H | to] -> coprime #|H| #|A| -> solvable H ->
'C_(|to)(A) / H = 'C_(|to / H)(A).
Proof.
move=> sHG nHA coHA solH; pose N := 'N_G(H).
have nsHN: H <| N by rewrite normal_subnorm.
have [sHN nHn] := andP nsHN.
have sNG: N \subset G by apply: subse... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | ext_coprime_quotient_cent | |
sol_coprime_Sylow_existsA G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
exists2 P : {group gT}, p.-Sylow(G) P & A \subset 'N(P).
Proof.
move=> solA nGA coGA; pose AG := A <*> G.
have nsG_AG: G <| AG by rewrite /normal joing_subr join_subG nGA normG.
have [sG_AG nG_AG]:= andP nsG_AG.
have [P sylP] := S... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | sol_coprime_Sylow_exists | |
sol_coprime_Sylow_transA G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
[transitive 'C_G(A), on [set P in 'Syl_p(G) | A \subset 'N(P)] | 'JG].
Proof.
move=> solA nGA coGA; pose AG := A <*> G; set FpA := finset _.
have nG_AG: AG \subset 'N(G) by rewrite join_subG nGA normG.
have [P sylP nPA] := sol_copr... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | sol_coprime_Sylow_trans | |
sol_coprime_Sylow_subsetA G X :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable A ->
X \subset G -> p.-group X -> A \subset 'N(X) ->
exists P : {group gT}, [/\ p.-Sylow(G) P, A \subset 'N(P) & X \subset P].
Proof.
move=> nGA coGA solA sXG pX nXA.
pose nAp (Q : {group gT}) := [&& p.-group Q, Q \subset G & A \subs... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice",
"From mathcomp Require Import fintype finset prime fingroup morphism",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import commutator center pgroup finmodule nilpotent... | solvable/hall.v | sol_coprime_Sylow_subset | |
section(gT : finGroupType) := GSection of {group gT} * {group gT}.
Delimit Scope section_scope with sec.
Bind Scope section_scope with section. | Inductive | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | section | |
mkSec(gT : finGroupType) (G1 G2 : {group gT}) := GSection (G1, G2).
Infix "/" := mkSec : section_scope. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | mkSec | |
pair_of_sectiongT (s : section gT) := let: GSection u := s in u. | Coercion | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | pair_of_section | |
quotient_of_sectiongT (s : section gT) : GroupSet.sort _ := s.1 / s.2. | Coercion | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | quotient_of_section | |
section_groupgT (s : section gT) : {group (coset_of s.2)} :=
Eval hnf in [group of s]. | Coercion | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | section_group | |
Definition_ := [isNew for (@pair_of_section gT)].
HB.instance Definition _ := [Finite of section gT by <:]. | HB.instance | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | Definition | |
section_group. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | section_group | |
section_isog:= [rel x y : section gT | x \isog y]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | section_isog | |
section_reprs := odflt (1 / 1)%sec (pick (section_isog ^~ s)). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | section_repr | |
mksreprG1 G2 := section_repr (mkSec G1 G2). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | mksrepr | |
section_reprPs : section_repr s \isog s.
Proof.
by rewrite /section_repr; case: pickP => //= /(_ s); rewrite isog_refl.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | section_reprP | |
section_repr_isogs1 s2 :
s1 \isog s2 -> section_repr s1 = section_repr s2.
Proof.
by move=> iso12; congr (odflt _ _); apply: eq_pick => s; apply: isog_transr.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | section_repr_isog | |
mkfactors(G : {group gT}) (s : seq {group gT}) :=
map section_repr (pairmap (@mkSec _) G s). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | mkfactors | |
compsG s := ((last G s) == 1%G) && compo.-series G s. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | comps | |
compsPG s :
reflect (last G s = 1%G /\ path [rel x y : gTg | maxnormal y x x] G s)
(comps G s).
Proof. by apply: (iffP andP) => [] [/eqP]. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | compsP | |
trivg_compsG s : comps G s -> (G :==: 1) = (s == [::]).
Proof.
case/andP=> ls cs; apply/eqP/eqP=> [G1 | s1]; last first.
by rewrite s1 /= in ls; apply/eqP.
by case: s {ls} cs => //= H s /andP[/maxgroupp]; rewrite G1 /proper sub1G andbF.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | trivg_comps | |
comps_consG H s : comps G (H :: s) -> comps H s.
Proof. by case/andP => /= ls /andP[_]; rewrite /comps ls. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | comps_cons | |
simple_compsPG s : comps G s -> reflect (s = [:: 1%G]) (simple G).
Proof.
move=> cs; apply: (iffP idP) => [|s1]; last first.
by rewrite s1 /comps eqxx /= andbT -simple_maxnormal in cs.
case: s cs => [/trivg_comps/eqP-> | H s]; first by case/simpleP; rewrite eqxx.
rewrite [comps _ _]andbCA /= => /andP[/maxgroupp maxH ... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | simple_compsP | |
exists_comps(G : gTg) : exists s, comps G s.
Proof.
elim: {G} #|G| {1 3}G (leqnn #|G|) => [G | n IHn G cG].
by rewrite leqNgt cardG_gt0.
have [sG | nsG] := boolP (simple G).
by exists [:: 1%G]; rewrite /comps eqxx /= -simple_maxnormal andbT.
have [-> | ntG] := eqVneq G 1%G; first by exists [::]; rewrite /comps eqx... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | exists_comps | |
JordanHolderUniqueness(G : gTg) (s1 s2 : seq gTg) :
comps G s1 -> comps G s2 -> perm_eq (mkfactors G s1) (mkfactors G s2).
Proof.
have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => /ltnSE-cG cs1 cs2.
have [G1 | ntG] := boolP (G :==: 1).
have -> : s1 = [::] by apply/eqP; rewrite -(trivg_comps cs1).
have -... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | JordanHolderUniqueness | |
gactsP(G : {set rT}) : reflect {acts A, on G | to} [acts A, on G | to].
Proof.
apply: (iffP idP) => [nGA x|nGA]; first exact: acts_act.
apply/subsetP=> a Aa /[!inE]; rewrite Aa.
by apply/subsetP=> x; rewrite inE nGA.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | gactsP | |
gactsM(N1 N2 : {set rT}) :
N1 \subset D -> N2 \subset D ->
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 * N2 | to].
Proof.
move=> sN1D sN2D aAN1 aAN2; apply/gactsP=> x Ax y.
apply/idP/idP; case/mulsgP=> y1 y2 N1y1 N2y2 e.
move: (actKin to Ax y); rewrite e; move<-.
rewrite gactM ?groupV ?(sub... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | gactsM | |
gactsI(N1 N2 : {set rT}) :
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 :&: N2 | to].
Proof.
move=> aAN1 aAN2.
apply/subsetP=> x Ax; rewrite !inE Ax /=; apply/subsetP=> y Ny /[1!inE].
case/setIP: Ny=> N1y N2y; rewrite inE ?astabs_act ?N1y ?N2y //.
- by move/subsetP: aAN2; move/(_ x Ax).
- by move/s... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | gactsI | |
gastabsP(S : {set rT}) (a : aT) :
a \in A -> reflect (forall x, (to x a \in S) = (x \in S)) (a \in 'N(S | to)).
Proof.
move=> Aa; apply: (iffP idP) => [nSa x|nSa]; first exact: astabs_act.
by rewrite !inE Aa; apply/subsetP=> x; rewrite inE nSa.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | gastabsP | |
qact_dom_doms(H : {group rT}) : H \subset D -> qact_dom to H \subset A.
Proof.
by move=> sHD; apply/subsetP=> x; rewrite qact_domE // inE; case/andP.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | qact_dom_doms | |
acts_qact_doms(H : {group rT}) :
H \subset D -> [acts A, on H | to] -> qact_dom to H :=: A.
Proof.
move=> sHD aH; apply/eqP; rewrite eqEsubset; apply/andP.
split; first exact: qact_dom_doms.
apply/subsetP=> x Ax; rewrite qact_domE //; apply/gastabsP=> //.
by move/gactsP: aH; move/(_ x Ax).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | acts_qact_doms | |
qacts_cosetpre(H : {group rT}) (K' : {group coset_of H}) :
H \subset D -> [acts A, on H | to] ->
[acts qact_dom to H, on K' | to / H] ->
[acts A, on coset H @*^-1 K' | to].
Proof.
move=> sHD aH aK'; apply/subsetP=> x Ax; move: (Ax) (subsetP aK').
rewrite -{1}(acts_qact_doms sHD aH) => qdx; move/(_ x qdx) => n... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | qacts_cosetpre | |
qacts_coset(H K : {group rT}) :
H \subset D -> [acts A, on K | to] ->
[acts qact_dom to H, on (coset H) @* K | to / H].
Proof.
move=> sHD aK.
apply/subsetP=> x qdx; rewrite inE qdx inE; apply/subsetP=> y.
case/morphimP=> z Nz Kz /= e; rewrite e inE qactE // imset_f // inE.
move/gactsP: aK; move/(_ x (subsetP (qac... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | qacts_coset | |
maxainv(B C : {set rT}) :=
[max C of H |
[&& (H <| B), ~~ (B \subset H) & [acts A, on H | to]]]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | maxainv | |
maxainv_norm: maxainv K N -> N <| K.
Proof. by move/maxgroupp; case/andP. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | maxainv_norm | |
maxainv_proper: maxainv K N -> N \proper K.
Proof.
by move/maxgroupp; case/andP; rewrite properE; move/normal_sub->; case/andP.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | maxainv_proper | |
maxainv_sub: maxainv K N -> N \subset K.
Proof. by move=> h; apply: proper_sub; apply: maxainv_proper. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | maxainv_sub | |
maxainv_ainvar: maxainv K N -> A \subset 'N(N | to).
Proof. by move/maxgroupp; case/and3P. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | maxainv_ainvar | |
maxainvS: maxainv K N -> N \subset K.
Proof. by move=> pNN; rewrite proper_sub // maxainv_proper. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | maxainvS | |
maxainv_exists: K :!=: 1 -> {N : {group rT} | maxainv K N}.
Proof.
move=> nt; apply: ex_maxgroup. exists [1 rT]%G.
rewrite /= normal1 subG1 nt /=.
apply/subsetP=> a Da; rewrite !inE Da /= sub1set !inE.
by rewrite /= -actmE // morph1 eqxx.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | maxainv_exists | |
maxainvM(G H K : {group rT}) :
H \subset D -> K \subset D -> maxainv G H -> maxainv G K ->
H :<>: K -> H * K = G.
Proof.
move: H K => N1 N2 sN1D sN2D pmN1 pmN2 neN12.
have cN12 : commute N1 N2.
apply: normC; apply: (subset_trans (maxainv_sub pmN1)).
by rewrite normal_norm ?maxainv_norm.
wlog nsN21 : G N1 N2 s... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | maxainvM | |
asimple(K : {set rT}) := maxainv K 1.
Implicit Types (H K : {group rT}) (s : seq {group rT}). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | asimple | |
asimplePK :
reflect [/\ K :!=: 1
& forall H, H <| K -> [acts A, on H | to] -> H :=: 1 \/ H :=: K]
(asimple K).
Proof.
apply: (iffP idP).
case/maxgroupP; rewrite normal1 /=; case/andP=> nsK1 aK H1.
rewrite eqEsubset negb_and nsK1 /=; split => // H nHK ha.
case eHK : (H :==: K); first by rig... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | asimpleP | |
acompsK s :=
((last K s) == 1%G) && path [rel x y : {group rT} | maxainv x y] K s. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | acomps | |
acompsPK s :
reflect (last K s = 1%G /\ path [rel x y : {group rT} | maxainv x y] K s)
(acomps K s).
Proof. by apply: (iffP andP); case; move/eqP. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | acompsP | |
trivg_acompsK s : acomps K s -> (K :==: 1) = (s == [::]).
Proof.
case/andP=> ls cs; apply/eqP/eqP; last first.
by move=> se; rewrite se /= in ls; apply/eqP.
move=> G1; case: s ls cs => // H s _ /=; case/andP; case/maxgroupP.
by rewrite G1 sub1G andbF.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | trivg_acomps | |
acomps_consK H s : acomps K (H :: s) -> acomps H s.
Proof. by case/andP => /= ls; case/andP=> _ p; rewrite /acomps ls. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | acomps_cons | |
asimple_acompsPK s : acomps K s -> reflect (s = [:: 1%G]) (asimple K).
Proof.
move=> cs; apply: (iffP idP); last first.
by move=> se; move: cs; rewrite se /=; case/andP=> /=; rewrite andbT.
case: s cs.
by rewrite /acomps /= andbT; move/eqP->; case/asimpleP; rewrite eqxx.
move=> H s cs sG; apply/eqP.
rewrite eqseq_c... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | asimple_acompsP | |
exists_acompsK : exists s, acomps K s.
Proof.
elim: {K} #|K| {1 3}K (leqnn #|K|) => [K | n Hi K cK].
by rewrite leqNgt cardG_gt0.
case/orP: (orbN (asimple K)) => [sK | nsK].
by exists [:: (1%G : {group rT})]; rewrite /acomps eqxx /= andbT.
case/orP: (orbN (K :==: 1))=> [tK | ntK].
by exists (Nil _); rewrite /acom... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | exists_acomps | |
maxainv_asimple_quo(G H : {group rT}) :
H \subset D -> maxainv to G H -> asimple (to / H) (G / H).
Proof.
move=> sHD /maxgroupP[/and3P[nHG pHG aH] Hmax].
apply/asimpleP; split; first by rewrite -subG1 quotient_sub1 ?normal_norm.
move=> K' nK'Q aK'.
have: (K' \proper (G / H)) || (G / H == K').
by rewrite properE eq... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | maxainv_asimple_quo | |
asimple_quo_maxainv(G H : {group rT}) :
H \subset D -> G \subset D -> [acts A, on G | to] -> [acts A, on H | to] ->
H <| G -> asimple (to / H) (G / H) ->
maxainv to G H.
Proof.
move=> sHD sGD aG aH nHG /asimpleP[ntQ maxQ]; apply/maxgroupP; split.
by rewrite nHG -quotient_sub1 ?normal_norm // subG1 ntQ.
move... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | asimple_quo_maxainv | |
asimpleI(N1 N2 : {group rT}) :
N2 \subset 'N(N1) -> N1 \subset D ->
[acts A, on N1 | to] -> [acts A, on N2 | to] ->
asimple (to / N1) (N2 / N1) ->
asimple (to / (N2 :&: N1)) (N2 / (N2 :&: N1)).
Proof.
move=> nN21 sN1D aN1 aN2 /asimpleP[ntQ1 max1].
have [f1 [f1e f1ker f1pre f1im]] := restrmP (coset_morphis... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | asimpleI | |
StrongJordanHolderUniqueness(G : {group rT}) (s1 s2 : seq {group rT}) :
G \subset D -> acomps to G s1 -> acomps to G s2 ->
perm_eq (mkfactors G s1) (mkfactors G s2).
Proof.
have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => cG hsD cs1 cs2.
case/orP: (orbN (G :==: 1)) => [tG | ntG].
have -> : s1 = [::] ... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action gseries"
] | solvable/jordanholder.v | StrongJordanHolderUniqueness | |
charsimpleA := [min A of G | G :!=: 1 & G \char A]. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | charsimple | |
FrattiniA := \bigcap_(G : {group gT} | maximal_eq G A) G. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Frattini | |
Frattini_groupA : {group gT} := Eval hnf in [group of Frattini A]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Frattini_group | |
FittingA := \big[dprod/1]_(p <- primes #|A|) 'O_p(A). | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting | |
Fitting_group_setG : group_set (Fitting G).
Proof.
suffices [F ->]: exists F : {group gT}, Fitting G = F by apply: groupP.
rewrite /Fitting; elim: primes (primes_uniq #|G|) => [_|p r IHr] /=.
by exists [1 gT]%G; rewrite big_nil.
case/andP=> rp /IHr[F defF]; rewrite big_cons defF.
suffices{IHr} /and3P[p'F sFG nFG]: p^... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_group_set | |
Fitting_groupG := group (Fitting_group_set G). | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_group | |
criticalA B :=
[/\ A \char B,
Frattini A \subset 'Z(A),
[~: B, A] \subset 'Z(A)
& 'C_B(A) = 'Z(A)]. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | critical | |
specialA := Frattini A = 'Z(A) /\ A^`(1) = 'Z(A). | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | special | |
extraspecialA := special A /\ prime #|'Z(A)|. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | extraspecial | |
SCNB := [set A : {group gT} | A <| B & 'C_B(A) == A]. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | SCN | |
SCN_atn B := [set A in SCN B | n <= 'r(A)]. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | SCN_at | |
p_maximal_normal: maximal M P -> M <| P.
Proof.
case/maxgroupP=> /andP[sMP sPM] maxM; rewrite /normal sMP.
have:= subsetIl P 'N(M); rewrite subEproper.
case/predU1P=> [/setIidPl-> // | /maxM/= SNM]; case/negP: sPM.
rewrite (nilpotent_sub_norm (pgroup_nil pP) sMP) //.
by rewrite SNM // subsetI sMP normG.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | p_maximal_normal | |
p_maximal_index: maximal M P -> #|P : M| = p.
Proof.
move=> maxM; have nM := p_maximal_normal maxM.
rewrite -card_quotient ?normal_norm //.
rewrite -(quotient_maximal _ nM) ?normal_refl // trivg_quotient in maxM.
case/maxgroupP: maxM; rewrite properEneq eq_sym sub1G andbT /=.
case/(pgroup_pdiv (quotient_pgroup M pP)) =... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | p_maximal_index | |
p_index_maximal: M \subset P -> prime #|P : M| -> maximal M P.
Proof.
move=> sMP /primeP[lt1PM pr_PM].
apply/maxgroupP; rewrite properEcard sMP -(Lagrange sMP).
rewrite -{1}(muln1 #|M|) ltn_pmul2l //; split=> // H sHP sMH.
apply/eqP; rewrite eq_sym eqEcard sMH.
case/orP: (pr_PM _ (indexSg sMH (proper_sub sHP))) => /eqP... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | p_index_maximal | |
Phi_subG : 'Phi(G) \subset G.
Proof. by rewrite bigcap_inf // /maximal_eq eqxx. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_sub | |
Phi_sub_maxG M : maximal M G -> 'Phi(G) \subset M.
Proof. by move=> maxM; rewrite bigcap_inf // /maximal_eq predU1r. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_sub_max | |
Phi_properG : G :!=: 1 -> 'Phi(G) \proper G.
Proof.
move/eqP; case/maximal_exists: (sub1G G) => [<- //| [M maxM _] _].
exact: sub_proper_trans (Phi_sub_max maxM) (maxgroupp maxM).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_proper | |
Phi_nongenG X : 'Phi(G) <*> X = G -> <<X>> = G.
Proof.
move=> defG; have: <<X>> \subset G by rewrite -{1}defG genS ?subsetUr.
case/maximal_exists=> //= [[M maxM]]; rewrite gen_subG => sXM.
case/andP: (maxgroupp maxM) => _ /negP[].
by rewrite -defG gen_subG subUset Phi_sub_max.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_nongen | |
Frattini_continuous(rT : finGroupType) G (f : {morphism G >-> rT}) :
f @* 'Phi(G) \subset 'Phi(f @* G).
Proof.
apply/bigcapsP=> M maxM; rewrite sub_morphim_pre ?Phi_sub // bigcap_inf //.
have {2}<-: f @*^-1 (f @* G) = G by rewrite morphimGK ?subsetIl.
by rewrite morphpre_maximal_eq ?maxM //; case/maximal_eqP: maxM.
Q... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Frattini_continuous | |
Frattini_igFun:= [igFun by Phi_sub & Frattini_continuous]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Frattini_igFun | |
Frattini_gFun:= [gFun by Frattini_continuous]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Frattini_gFun | |
Phi_charG : 'Phi(G) \char G.
Proof. exact: gFchar. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_char | |
Phi_normalG : 'Phi(G) <| G.
Proof. exact: gFnormal. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_normal | |
injm_PhirT D G (f : {morphism D >-> rT}) :
'injm f -> G \subset D -> f @* 'Phi(G) = 'Phi(f @* G).
Proof. exact: injmF. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | injm_Phi | |
isog_PhirT G (H : {group rT}) : G \isog H -> 'Phi(G) \isog 'Phi(H).
Proof. exact: gFisog. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | isog_Phi | |
PhiJG x : 'Phi(G :^ x) = 'Phi(G) :^ x.
Proof.
rewrite -{1}(setIid G) -(setIidPr (Phi_sub G)) -!morphim_conj.
by rewrite injm_Phi ?injm_conj.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | PhiJ | |
Phi_quotient_idG : 'Phi (G / 'Phi(G)) = 1.
Proof.
apply/trivgP; rewrite -cosetpreSK cosetpre1 /=; apply/bigcapsP=> M maxM.
have nPhi := Phi_normal G; have nPhiM: 'Phi(G) <| M.
by apply: normalS nPhi; [apply: bigcap_inf | case/maximal_eqP: maxM].
by rewrite sub_cosetpre_quo ?bigcap_inf // quotient_maximal_eq.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_quotient_id | |
Phi_quotient_cyclicG : cyclic (G / 'Phi(G)) -> cyclic G.
Proof.
case/cyclicP=> /= Px; case: (cosetP Px) => x nPx ->{Px} defG.
apply/cyclicP; exists x; symmetry; apply: Phi_nongen.
rewrite -joing_idr norm_joinEr -?quotientK ?cycle_subG //.
by rewrite /quotient morphim_cycle //= -defG quotientGK ?Phi_normal.
Qed.
Variabl... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_quotient_cyclic | |
trivg_Phi: p.-group P -> ('Phi(P) == 1) = p.-abelem P.
Proof.
move=> pP; case: (eqsVneq P 1) => [P1 | ntP].
by rewrite P1 abelem1 -subG1 -P1 Phi_sub.
have [p_pr _ _] := pgroup_pdiv pP ntP.
apply/eqP/idP=> [trPhi | abP].
apply/abelemP=> //; split=> [|x Px].
apply/commG1P/trivgP; rewrite -trPhi.
apply/bigcaps... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | trivg_Phi | |
Phi_quotient_abelem: p.-abelem (P / 'Phi(P)).
Proof. by rewrite -trivg_Phi ?morphim_pgroup //= Phi_quotient_id. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_quotient_abelem | |
Phi_joing: 'Phi(P) = P^`(1) <*> 'Mho^1(P).
Proof.
have [sPhiP nPhiP] := andP (Phi_normal P).
apply/eqP; rewrite eqEsubset join_subG.
case: (eqsVneq P 1) => [-> | ntP] in sPhiP *.
by rewrite /= (trivgP sPhiP) sub1G der_subS Mho_sub.
have [p_pr _ _] := pgroup_pdiv pP ntP.
have [abP x1P] := abelemP p_pr Phi_quotient_abe... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_joing | |
Phi_Mho: abelian P -> 'Phi(P) = 'Mho^1(P).
Proof. by move=> cPP; rewrite Phi_joing (derG1P cPP) joing1G. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_Mho | |
PhiSG H : p.-group H -> G \subset H -> 'Phi(G) \subset 'Phi(H).
Proof.
move=> pH sGH; rewrite (Phi_joing pH) (Phi_joing (pgroupS sGH pH)).
by rewrite genS // setUSS ?dergS ?MhoS.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | PhiS | |
morphim_PhirT P D (f : {morphism D >-> rT}) :
p.-group P -> P \subset D -> f @* 'Phi(P) = 'Phi(f @* P).
Proof.
move=> pP sPD; rewrite !(@Phi_joing _ p) ?morphim_pgroup //.
rewrite morphim_gen ?subUset ?gFsub_trans // morphimU -joingE.
by rewrite morphimR ?morphim_Mho.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | morphim_Phi | |
quotient_PhiP H :
p.-group P -> P \subset 'N(H) -> 'Phi(P) / H = 'Phi(P / H).
Proof. exact: morphim_Phi. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | quotient_Phi | |
Phi_minG H :
p.-group G -> G \subset 'N(H) -> p.-abelem (G / H) -> 'Phi(G) \subset H.
Proof.
move=> pG nHG; rewrite -trivg_Phi ?quotient_pgroup // -subG1 /=.
by rewrite -(quotient_Phi pG) ?quotient_sub1 // gFsub_trans.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_min | |
Phi_cprodG H K :
p.-group G -> H \* K = G -> 'Phi(H) \* 'Phi(K) = 'Phi(G).
Proof.
move=> pG defG; have [_ /mulG_sub[sHG sKG] cHK] := cprodP defG.
rewrite cprodEY /=; last by rewrite (centSS (Phi_sub _) (Phi_sub _)).
rewrite !(Phi_joing (pgroupS _ pG)) //=.
have /cprodP[_ <- /cent_joinEr <-] := der_cprod 1 defG.
have ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_cprod | |
Phi_mulgH K :
p.-group H -> p.-group K -> K \subset 'C(H) ->
'Phi(H * K) = 'Phi(H) * 'Phi(K).
Proof.
move=> pH pK cHK; have defHK := cprodEY cHK.
have [|_ ->] /= := cprodP (Phi_cprod _ defHK); rewrite cent_joinEr //.
by rewrite pgroupM pH.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Phi_mulg |
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