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 |
|---|---|---|---|---|---|---|
morphim_pSylowp G P :
P \subset D -> p.-Sylow(G) P -> p.-Sylow(f @* G) (f @* P).
Proof. exact: morphim_pHall. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_pSylow
| |
morphim_p_groupP : p_group P -> p_group (f @* P).
Proof. by move/morphim_pgroup; apply: pgroup_p. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_p_group
| |
morphim_SylowG P : P \subset D -> Sylow G P -> Sylow (f @* G) (f @* P).
Proof.
by move=> sPD /andP[pP hallP]; rewrite /Sylow morphim_p_group // morphim_Hall.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_Sylow
| |
morph_p_eltpi x : x \in D -> pi.-elt x -> pi.-elt (f x).
Proof. by move=> Dx; apply: pnat_dvd; apply: morph_order. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morph_p_elt
| |
morph_consttpi x : x \in D -> f x.`_pi = (f x).`_pi.
Proof.
move=> Dx; rewrite -{2}(consttC pi x) morphM ?groupX //.
rewrite consttM; last by rewrite !morphX //; apply: commuteX2.
have: pi.-elt (f x.`_pi) by rewrite morph_p_elt ?groupX ?p_elt_constt //.
have: pi^'.-elt (f x.`_pi^') by rewrite morph_p_elt ?groupX ?p_elt_constt //.
by move/constt1P->; move/constt_p_elt->; rewrite mulg1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morph_constt
| |
quotient_pgroup: pi.-group (K / H). Proof. exact: morphim_pgroup. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
quotient_pgroup
| |
quotient_pHall:
K \subset 'N(H) -> pi.-Hall(G) K -> pi.-Hall(G / H) (K / H).
Proof. exact: morphim_pHall. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
quotient_pHall
| |
quotient_odd: odd #|K| -> odd #|K / H|. Proof. exact: morphim_odd. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
quotient_odd
| |
pquotient_pgroup: G \subset 'N(K) -> pi.-group (G / K) = pi.-group G.
Proof. by move=> nKG; rewrite pmorphim_pgroup ?ker_coset. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pquotient_pgroup
| |
pquotient_pHall:
K <| G -> K <| H -> pi.-Hall(G / K) (H / K) = pi.-Hall(G) H.
Proof.
case/andP=> sKG nKG; case/andP=> sKH nKH.
by rewrite pmorphim_pHall // ker_coset /psubgroup subsetI sKH sKG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pquotient_pHall
| |
ltn_log_quotient:
p.-group G -> H :!=: 1 -> H \subset G -> logn p #|G / H| < logn p #|G|.
Proof.
move=> pG ntH sHG; apply: contraLR (ltn_quotient ntH sHG); rewrite -!leqNgt.
rewrite {2}(card_pgroup pG) {2}(card_pgroup (morphim_pgroup _ pG)).
by case: (posnP p) => [-> //|]; apply: leq_pexp2l.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
ltn_log_quotient
| |
logn_quotient_cent_cyclic_pgroup:
p.-group C -> cyclic C -> logn p #|G / 'C_G(C)| <= (logn p #|C|).-1.
Proof.
move=> pC cycC; have [-> | ntC] := eqsVneq C 1.
by rewrite cent1T setIT trivg_quotient cards1 logn1.
have [p_pr _ [e oC]] := pgroup_pdiv pC ntC.
rewrite -ker_conj_aut (card_isog (first_isog_loc _ _)) //.
apply: leq_trans (dvdn_leq_log _ _ (cardSg (Aut_conj_aut _ _))) _ => //.
rewrite card_Aut_cyclic // oC totient_pfactor //= logn_Gauss ?pfactorK //.
by rewrite prime_coprime // gtnNdvd // -(subnKC (prime_gt1 p_pr)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
logn_quotient_cent_cyclic_pgroup
| |
p'group_quotient_cent_prime:
prime p -> #|C| %| p -> p^'.-group (G / 'C_G(C)).
Proof.
move=> p_pr pC; have pgC: p.-group C := pnat_dvd pC (pnat_id p_pr).
have [_ dv_p] := primeP p_pr; case/pred2P: {dv_p pC}(dv_p _ pC) => [|pC].
by move/card1_trivg->; rewrite cent1T setIT trivg_quotient pgroup1.
have le_oGC := logn_quotient_cent_cyclic_pgroup pgC.
rewrite /pgroup -partn_eq1 ?cardG_gt0 // -dvdn1 p_part pfactor_dvdn // logn1.
by rewrite (leq_trans (le_oGC _)) ?prime_cyclic // pC ?(pfactorK 1).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p'group_quotient_cent_prime
| |
pcore:= \bigcap_(G | [max G | pi.-subgroup(A) G]) G.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore
| |
pcore_group: {group gT} := Eval hnf in [group of pcore].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_group
| |
pcore_modpi B := coset B @*^-1 'O_pi(A / B).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_mod
| |
pcore_mod_grouppi B : {group gT} :=
Eval hnf in [group of pcore_mod pi B].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_mod_group
| |
pseries:= foldr pcore_mod 1 (rev pis).
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries
| |
pseries_group_set: group_set pseries.
Proof. by rewrite /pseries; case: rev => [|pi1 pi1']; apply: groupP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_group_set
| |
pseries_group: {group gT} := group pseries_group_set.
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_group
| |
pcore_psubgroupG : pi.-subgroup(G) 'O_pi(G).
Proof.
have [M maxM _]: {M | [max M | pi.-subgroup(G) M] & 1%G \subset M}.
by apply: maxgroup_exists; rewrite /psubgroup sub1G pgroup1.
have sOM: 'O_pi(G) \subset M by apply: bigcap_inf.
have /andP[piM sMG] := maxgroupp maxM.
by rewrite /psubgroup (pgroupS sOM) // (subset_trans sOM).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_psubgroup
| |
pcore_pgroupG : pi.-group 'O_pi(G).
Proof. by case/andP: (pcore_psubgroup G). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_pgroup
| |
pcore_subG : 'O_pi(G) \subset G.
Proof. by case/andP: (pcore_psubgroup G). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_sub
| |
pcore_sub_HallG H : pi.-Hall(G) H -> 'O_pi(G) \subset H.
Proof. by move/Hall_max=> maxH; apply: bigcap_inf. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_sub_Hall
| |
pcore_maxG H : pi.-group H -> H <| G -> H \subset 'O_pi(G).
Proof.
move=> piH nHG; apply/bigcapsP=> M maxM.
exact: normal_sub_max_pgroup piH nHG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_max
| |
pcore_pgroup_idG : pi.-group G -> 'O_pi(G) = G.
Proof. by move=> piG; apply/eqP; rewrite eqEsubset pcore_sub pcore_max. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_pgroup_id
| |
pcore_normalG : 'O_pi(G) <| G.
Proof.
rewrite /(_ <| G) pcore_sub; apply/subsetP=> x Gx.
rewrite inE; apply/bigcapsP=> M maxM; rewrite sub_conjg.
by apply: bigcap_inf; apply: max_pgroupJ; rewrite ?groupV.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_normal
| |
normal_Hall_pcoreH G : pi.-Hall(G) H -> H <| G -> 'O_pi(G) = H.
Proof.
move=> hallH nHG; apply/eqP.
rewrite eqEsubset (sub_normal_Hall hallH) ?pcore_sub ?pcore_pgroup //=.
by rewrite pcore_max //= (pHall_pgroup hallH).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
normal_Hall_pcore
| |
eq_Hall_pcoreG H :
pi.-Hall(G) 'O_pi(G) -> pi.-Hall(G) H -> H :=: 'O_pi(G).
Proof.
move=> hallGpi hallH.
exact: uniq_normal_Hall (pcore_normal G) (Hall_max hallH).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
eq_Hall_pcore
| |
sub_Hall_pcoreG K :
pi.-Hall(G) 'O_pi(G) -> K \subset G -> (K \subset 'O_pi(G)) = pi.-group K.
Proof. by move=> hallGpi; apply: sub_normal_Hall (pcore_normal G). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sub_Hall_pcore
| |
mem_Hall_pcoreG x :
pi.-Hall(G) 'O_pi(G) -> x \in G -> (x \in 'O_pi(G)) = pi.-elt x.
Proof. by move=> hallGpi; apply: mem_normal_Hall (pcore_normal G). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
mem_Hall_pcore
| |
sdprod_Hall_pcorePH G :
pi.-Hall(G) 'O_pi(G) -> reflect ('O_pi(G) ><| H = G) (pi^'.-Hall(G) H).
Proof.
move=> hallGpi; rewrite -(compl_pHall H hallGpi) complgC.
exact: sdprod_normal_complP (pcore_normal G).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sdprod_Hall_pcoreP
| |
sdprod_pcore_HallPH G :
pi^'.-Hall(G) H -> reflect ('O_pi(G) ><| H = G) (pi.-Hall(G) 'O_pi(G)).
Proof. exact: sdprod_normal_p'HallP (pcore_normal G). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sdprod_pcore_HallP
| |
pcoreJG x : 'O_pi(G :^ x) = 'O_pi(G) :^ x.
Proof.
apply/eqP; rewrite eqEsubset -sub_conjgV.
rewrite !pcore_max ?pgroupJ ?pcore_pgroup ?normalJ ?pcore_normal //.
by rewrite -(normalJ _ _ x) conjsgKV pcore_normal.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcoreJ
| |
morphim_pcorepi : GFunctor.pcontinuous (@pcore pi).
Proof.
move=> gT rT D G f; apply/bigcapsP=> M /normal_sub_max_pgroup; apply.
by rewrite morphim_pgroup ?pcore_pgroup.
by apply: morphim_normal; apply: pcore_normal.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_pcore
| |
pcoreSpi gT (G H : {group gT}) :
H \subset G -> H :&: 'O_pi(G) \subset 'O_pi(H).
Proof.
move=> sHG; rewrite -{2}(setIidPl sHG).
by do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; apply: morphim_pcore.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcoreS
| |
pcore_igFunpi := [igFun by pcore_sub pi & morphim_pcore pi].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_igFun
| |
pcore_gFunpi := [gFun by morphim_pcore pi].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_gFun
| |
pcore_pgFunpi := [pgFun by morphim_pcore pi].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_pgFun
| |
pcore_charpi gT (G : {group gT}) : 'O_pi(G) \char G.
Proof. exact: gFchar. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_char
| |
pcore_mod_subpi gT (G : {group gT}) : pcore_mod G pi (F _ G) \subset G.
Proof.
by rewrite sub_morphpre_im ?gFsub_trans ?morphimS ?gFnorm //= ker_coset gFsub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_mod_sub
| |
quotient_pcore_modpi gT (G : {group gT}) (B : {set gT}) :
pcore_mod G pi B / B = 'O_pi(G / B).
Proof. exact/morphpreK/gFsub_trans/morphim_sub. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
quotient_pcore_mod
| |
morphim_pcore_modpi gT rT (D G : {group gT}) (f : {morphism D >-> rT}) :
f @* pcore_mod G pi (F _ G) \subset pcore_mod (f @* G) pi (F _ (f @* G)).
Proof.
have sDF: D :&: G \subset 'dom (coset (F _ G)).
by rewrite setIC subIset ?gFnorm.
have sDFf: D :&: G \subset 'dom (coset (F _ (f @* G)) \o f).
by rewrite -sub_morphim_pre ?subsetIl // morphimIdom gFnorm.
pose K := 'ker (restrm sDFf (coset (F _ (f @* G)) \o f)).
have sFK: 'ker (restrm sDF (coset (F _ G))) \subset K.
rewrite /K !ker_restrm ker_comp /= subsetI subsetIl /= -setIA.
rewrite -sub_morphim_pre ?subsetIl //.
by rewrite morphimIdom !ker_coset (setIidPr _) ?pmorphimF ?gFsub.
have sOF := pcore_sub pi (G / F _ G); have sDD: D :&: G \subset D :&: G by [].
rewrite -sub_morphim_pre -?quotientE; last first.
by apply: subset_trans (gFnorm F _); rewrite morphimS ?pcore_mod_sub.
suffices im_fact (H : {group gT}) : F _ G \subset H -> H \subset G ->
factm sFK sDD @* (H / F _ G) = f @* H / F _ (f @* G).
- rewrite -2?im_fact ?pcore_mod_sub ?gFsub //;
try by rewrite -{1}[F _ G]ker_coset morphpreS ?sub1G.
by rewrite quotient_pcore_mod morphim_pcore.
move=> sFH sHG; rewrite -(morphimIdom _ (H / _)) /= {2}morphim_restrm setIid.
rewrite -morphimIG ?ker_coset //.
rewrite -(morphim_restrm sDF) morphim_factm morphim_restrm.
by rewrite morphim_comp -quotientE -setIA morphimIdom (setIidPr _).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_pcore_mod
| |
pcore_mod_respi gT rT (D : {group gT}) (f : {morphism D >-> rT}) :
f @* pcore_mod D pi (F _ D) \subset pcore_mod (f @* D) pi (F _ (f @* D)).
Proof. exact: morphim_pcore_mod. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_mod_res
| |
pcore_mod1pi gT (G : {group gT}) : pcore_mod G pi 1 = 'O_pi(G).
Proof.
rewrite /pcore_mod; have inj1 := coset1_injm gT; rewrite -injmF ?norms1 //.
by rewrite -(morphim_invmE inj1) morphim_invm ?norms1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_mod1
| |
pseries_rconspi pis gT (A : {set gT}) :
pseries (rcons pis pi) A = pcore_mod A pi (pseries pis A).
Proof. by rewrite /pseries rev_rcons. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_rcons
| |
pseries_subfunpis :
GFunctor.closed (@pseries pis) /\ GFunctor.pcontinuous (@pseries pis).
Proof.
elim/last_ind: pis => [|pis pi [sFpi fFpi]].
by split=> [gT G | gT rT D G f]; rewrite (sub1G, morphim1).
pose fF := [gFun by fFpi : GFunctor.continuous [igFun by sFpi & fFpi]].
pose F := [pgFun by fFpi : GFunctor.hereditary fF].
split=> [gT G | gT rT D G f]; rewrite !pseries_rcons ?(pcore_mod_sub F) //.
exact: (morphim_pcore_mod F).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_subfun
| |
pseries_subpis : GFunctor.closed (@pseries pis).
Proof. by case: (pseries_subfun pis). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_sub
| |
morphim_pseriespis : GFunctor.pcontinuous (@pseries pis).
Proof. by case: (pseries_subfun pis). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_pseries
| |
pseriesSpis : GFunctor.hereditary (@pseries pis).
Proof. exact: (morphim_pseries pis). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseriesS
| |
pseries_igFunpis := [igFun by pseries_sub pis & morphim_pseries pis].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_igFun
| |
pseries_gFunpis := [gFun by morphim_pseries pis].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_gFun
| |
pseries_pgFunpis := [pgFun by morphim_pseries pis].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_pgFun
| |
pseries_charpis gT (G : {group gT}) : pseries pis G \char G.
Proof. exact: gFchar. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_char
| |
pseries_normalpis gT (G : {group gT}) : pseries pis G <| G.
Proof. exact: gFnormal. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_normal
| |
pseriesJpis gT (G : {group gT}) x :
pseries pis (G :^ x) = pseries pis G :^ x.
Proof.
rewrite -{1}(setIid G) -morphim_conj -(injmF _ (injm_conj G x)) //=.
by rewrite morphim_conj (setIidPr (pseries_sub _ _)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseriesJ
| |
pseries1pi gT (G : {group gT}) : 'O_{pi}(G) = 'O_pi(G).
Proof. exact: pcore_mod1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries1
| |
pseries_poppi pis gT (G : {group gT}) :
'O_pi(G) = 1 -> pseries (pi :: pis) G = pseries pis G.
Proof.
by move=> OG1; rewrite /pseries rev_cons -cats1 foldr_cat /= pcore_mod1 OG1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_pop
| |
pseries_pop2pi1 pi2 gT (G : {group gT}) :
'O_pi1(G) = 1 -> 'O_{pi1, pi2}(G) = 'O_pi2(G).
Proof. by move/pseries_pop->; apply: pseries1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_pop2
| |
pseries_sub_catlpi1s pi2s gT (G : {group gT}) :
pseries pi1s G \subset pseries (pi1s ++ pi2s) G.
Proof.
elim/last_ind: pi2s => [|pi pis IHpi]; rewrite ?cats0 // -rcons_cat.
by rewrite pseries_rcons; apply: subset_trans IHpi _; rewrite sub_cosetpre.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_sub_catl
| |
quotient_pseriespis pi gT (G : {group gT}) :
pseries (rcons pis pi) G / pseries pis G = 'O_pi(G / pseries pis G).
Proof. by rewrite pseries_rcons quotient_pcore_mod. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
quotient_pseries
| |
pseries_norm2pi1s pi2s gT (G : {group gT}) :
pseries pi2s G \subset 'N(pseries pi1s G).
Proof. by rewrite gFsub_trans ?gFnorm. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_norm2
| |
pseries_sub_catrpi1s pi2s gT (G : {group gT}) :
pseries pi2s G \subset pseries (pi1s ++ pi2s) G.
Proof.
elim: pi1s => //= pi1 pi1s /subset_trans; apply.
elim/last_ind: {pi1s pi2s}(_ ++ _) => [|pis pi IHpi]; first exact: sub1G.
rewrite -rcons_cons (pseries_rcons _ (pi1 :: pis)).
rewrite -sub_morphim_pre ?pseries_norm2 //.
apply: pcore_max; last by rewrite morphim_normal ?pseries_normal.
have: pi.-group (pseries (rcons pis pi) G / pseries pis G).
by rewrite quotient_pseries pcore_pgroup.
by apply: pnat_dvd; rewrite !card_quotient ?pseries_norm2 // indexgS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_sub_catr
| |
quotient_pseries2pi1 pi2 gT (G : {group gT}) :
'O_{pi1, pi2}(G) / 'O_pi1(G) = 'O_pi2(G / 'O_pi1(G)).
Proof. by rewrite -pseries1 -quotient_pseries. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
quotient_pseries2
| |
quotient_pseries_catpi1s pi2s gT (G : {group gT}) :
pseries (pi1s ++ pi2s) G / pseries pi1s G
= pseries pi2s (G / pseries pi1s G).
Proof.
elim/last_ind: pi2s => [|pi2s pi IHpi]; first by rewrite cats0 trivg_quotient.
have psN := pseries_normal _ G; set K := pseries _ G.
case: (third_isom (pseries_sub_catl pi1s pi2s G) (psN _)) => //= f inj_f im_f.
have nH2H: pseries pi2s (G / K) <| pseries (pi1s ++ rcons pi2s pi) G / K.
rewrite -IHpi morphim_normal // -cats1 catA.
by apply/andP; rewrite pseries_sub_catl pseries_norm2.
apply: (quotient_inj nH2H).
by apply/andP; rewrite /= -cats1 pseries_sub_catl pseries_norm2.
rewrite /= quotient_pseries /= -IHpi -rcons_cat.
rewrite -[G / _ / _](morphim_invm inj_f) //= {2}im_f //.
rewrite -(@injmF [igFun of @pcore pi]) /= ?injm_invm ?im_f // -quotient_pseries.
by rewrite -im_f ?morphim_invm ?morphimS ?normal_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
quotient_pseries_cat
| |
pseries_catl_idpi1s pi2s gT (G : {group gT}) :
pseries pi1s (pseries (pi1s ++ pi2s) G) = pseries pi1s G.
Proof.
elim/last_ind: pi1s => [//|pi1s pi IHpi] in pi2s *.
apply: (@quotient_inj _ (pseries_group pi1s G)).
- rewrite /= -(IHpi (pi :: pi2s)) cat_rcons /(_ <| _) pseries_norm2.
by rewrite -cats1 pseries_sub_catl.
- by rewrite /= /(_ <| _) pseries_norm2 -cats1 pseries_sub_catl.
rewrite /= cat_rcons -(IHpi (pi :: pi2s)) {1}quotient_pseries IHpi.
apply/eqP; rewrite quotient_pseries eqEsubset !pcore_max ?pcore_pgroup //=.
rewrite -quotient_pseries morphim_normal // /(_ <| _) pseries_norm2.
by rewrite -cat_rcons pseries_sub_catl.
by rewrite gFnormal_trans ?quotient_normal ?gFnormal.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_catl_id
| |
pseries_char_catlpi1s pi2s gT (G : {group gT}) :
pseries pi1s G \char pseries (pi1s ++ pi2s) G.
Proof. by rewrite -(pseries_catl_id pi1s pi2s G) pseries_char. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_char_catl
| |
pseries_catr_idpi1s pi2s gT (G : {group gT}) :
pseries pi2s (pseries (pi1s ++ pi2s) G) = pseries pi2s G.
Proof.
elim/last_ind: pi2s => [//|pi2s pi IHpi] in G *.
have Epis: pseries pi2s (pseries (pi1s ++ rcons pi2s pi) G) = pseries pi2s G.
by rewrite -cats1 catA -[RHS]IHpi -[LHS]IHpi /= [pseries (_ ++ _) _]pseries_catl_id.
apply: (@quotient_inj _ (pseries_group pi2s G)).
- by rewrite /= -Epis /(_ <| _) pseries_norm2 -cats1 pseries_sub_catl.
- by rewrite /= /(_ <| _) pseries_norm2 -cats1 pseries_sub_catl.
rewrite /= -Epis {1}quotient_pseries Epis quotient_pseries.
apply/eqP; rewrite eqEsubset !pcore_max ?pcore_pgroup //=.
rewrite -quotient_pseries morphim_normal // /(_ <| _) pseries_norm2.
by rewrite pseries_sub_catr.
by rewrite gFnormal_trans ?morphim_normal ?gFnormal.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_catr_id
| |
pseries_char_catrpi1s pi2s gT (G : {group gT}) :
pseries pi2s G \char pseries (pi1s ++ pi2s) G.
Proof. by rewrite -(pseries_catr_id pi1s pi2s G) pseries_char. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_char_catr
| |
pcore_modppi gT (G H : {group gT}) :
H <| G -> pi.-group H -> pcore_mod G pi H = 'O_pi(G).
Proof.
move=> nsHG piH; have nHG := normal_norm nsHG; apply/eqP.
rewrite eqEsubset andbC -sub_morphim_pre ?(gFsub_trans, morphim_pcore) //=.
rewrite -[G in 'O_pi(G)](quotientGK nsHG) pcore_max //.
by rewrite -(pquotient_pgroup piH) ?subsetIl // cosetpreK pcore_pgroup.
by rewrite morphpre_normal ?gFnormal ?gFsub_trans ?morphim_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_modp
| |
pquotient_pcorepi gT (G H : {group gT}) :
H <| G -> pi.-group H -> 'O_pi(G / H) = 'O_pi(G) / H.
Proof. by move=> nsHG piH; rewrite -quotient_pcore_mod pcore_modp. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pquotient_pcore
| |
trivg_pcore_quotientpi gT (G : {group gT}) : 'O_pi(G / 'O_pi(G)) = 1.
Proof. by rewrite pquotient_pcore ?gFnormal ?pcore_pgroup ?trivg_quotient. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
trivg_pcore_quotient
| |
pseries_rcons_idpis pi gT (G : {group gT}) :
pseries (rcons (rcons pis pi) pi) G = pseries (rcons pis pi) G.
Proof.
apply/eqP; rewrite -!cats1 eqEsubset pseries_sub_catl andbT -catA.
rewrite -(quotientSGK _ (pseries_sub_catl _ _ _)) ?pseries_norm2 //.
rewrite !quotient_pseries_cat -quotient_sub1 ?pseries_norm2 //.
by rewrite quotient_pseries_cat /= !pseries1 trivg_pcore_quotient.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pseries_rcons_id
| |
sub_in_pcorepi rho G :
{in \pi(G), {subset pi <= rho}} -> 'O_pi(G) \subset 'O_rho(G).
Proof.
move=> pi_sub_rho; rewrite pcore_max ?pcore_normal //.
apply: sub_in_pnat (pcore_pgroup _ _) => p.
by move/(piSg (pcore_sub _ _)); apply: pi_sub_rho.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sub_in_pcore
| |
sub_pcorepi rho G : {subset pi <= rho} -> 'O_pi(G) \subset 'O_rho(G).
Proof. by move=> pi_sub_rho; apply: sub_in_pcore (in1W pi_sub_rho). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sub_pcore
| |
eq_in_pcorepi rho G : {in \pi(G), pi =i rho} -> 'O_pi(G) = 'O_rho(G).
Proof.
move=> eq_pi_rho; apply/eqP; rewrite eqEsubset.
by rewrite !sub_in_pcore // => p /eq_pi_rho->.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
eq_in_pcore
| |
eq_pcorepi rho G : pi =i rho -> 'O_pi(G) = 'O_rho(G).
Proof. by move=> eq_pi_rho; apply: eq_in_pcore (in1W eq_pi_rho). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
eq_pcore
| |
pcoreNKpi G : 'O_pi^'^'(G) = 'O_pi(G).
Proof. by apply: eq_pcore; apply: negnK. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcoreNK
| |
eq_p'corepi rho G : pi =i rho -> 'O_pi^'(G) = 'O_rho^'(G).
Proof. by move/eq_negn; apply: eq_pcore. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
eq_p'core
| |
sdprod_Hall_p'corePpi H G :
pi^'.-Hall(G) 'O_pi^'(G) -> reflect ('O_pi^'(G) ><| H = G) (pi.-Hall(G) H).
Proof. by rewrite -(pHallNK pi G H); apply: sdprod_Hall_pcoreP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sdprod_Hall_p'coreP
| |
sdprod_p'core_HallPpi H G :
pi.-Hall(G) H -> reflect ('O_pi^'(G) ><| H = G) (pi^'.-Hall(G) 'O_pi^'(G)).
Proof. by rewrite -(pHallNK pi G H); apply: sdprod_pcore_HallP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sdprod_p'core_HallP
| |
pcoreIpi rho G : 'O_[predI pi & rho](G) = 'O_pi('O_rho(G)).
Proof.
apply/eqP; rewrite eqEsubset !pcore_max //.
- rewrite /pgroup pnatI -!pgroupE.
by rewrite pcore_pgroup (pgroupS (pcore_sub pi _))// pcore_pgroup.
- by rewrite !gFnormal_trans.
- by apply: sub_pgroup (pcore_pgroup _ _) => p /andP[].
apply/andP; split; first by apply: sub_pcore => p /andP[].
by rewrite gFnorm_trans ?normsG ?gFsub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcoreI
| |
bigcap_p'corepi G :
G :&: \bigcap_(p < #|G|.+1 | (p : nat) \in pi) 'O_p^'(G) = 'O_pi^'(G).
Proof.
apply/eqP; rewrite eqEsubset subsetI pcore_sub pcore_max /=.
- by apply/bigcapsP=> p pi_p; apply: sub_pcore => r; apply: contraNneq => ->.
- apply/pgroupP=> q q_pr qGpi'; apply: contraL (eqxx q) => /= pi_q.
apply: (pgroupP (pcore_pgroup q^' G)) => //.
have qG: q %| #|G| by rewrite (dvdn_trans qGpi') // cardSg ?subsetIl.
have ltqG: q < #|G|.+1 by rewrite ltnS dvdn_leq.
rewrite (dvdn_trans qGpi') ?cardSg ?subIset //= orbC.
by rewrite (bigcap_inf (Ordinal ltqG)).
rewrite /normal subsetIl normsI ?normG // norms_bigcap //.
by apply/bigcapsP => p _; apply: gFnorm.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
bigcap_p'core
| |
coprime_pcoreC(rT : finGroupType) pi G (R : {group rT}) :
coprime #|'O_pi(G)| #|'O_pi^'(R)|.
Proof. exact: pnat_coprime (pcore_pgroup _ _) (pcore_pgroup _ _). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
coprime_pcoreC
| |
TI_pcoreCpi G H : 'O_pi(G) :&: 'O_pi^'(H) = 1.
Proof. by rewrite coprime_TIg ?coprime_pcoreC. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
TI_pcoreC
| |
pcore_setI_normalpi G H : H <| G -> 'O_pi(G) :&: H = 'O_pi(H).
Proof.
move=> nsHG; apply/eqP; rewrite eqEsubset subsetI pcore_sub setIC.
rewrite !pcore_max ?(pgroupS (subsetIr H _)) ?pcore_pgroup ?gFnormal_trans //=.
by rewrite norm_normalI ?gFnorm_trans ?normsG ?normal_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pcore_setI_normal
| |
injm_pgrouppi A : A \subset D -> pi.-group (f @* A) = pi.-group A.
Proof. by move=> sAD; rewrite /pgroup card_injm. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
injm_pgroup
| |
injm_peltpi x : x \in D -> pi.-elt (f x) = pi.-elt x.
Proof. by move=> Dx; rewrite /p_elt order_injm. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
injm_pelt
| |
injm_pHallpi G H :
G \subset D -> H \subset D -> pi.-Hall(f @* G) (f @* H) = pi.-Hall(G) H.
Proof. by move=> sGD sGH; rewrite !pHallE injmSK ?card_injm. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
injm_pHall
| |
injm_pcorepi G : G \subset D -> f @* 'O_pi(G) = 'O_pi(f @* G).
Proof. exact: injmF. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
injm_pcore
| |
injm_pseriespis G :
G \subset D -> f @* pseries pis G = pseries pis (f @* G).
Proof. exact: injmF. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
injm_pseries
| |
isog_pgrouppi : G \isog H -> pi.-group G = pi.-group H.
Proof. by move=> isoGH; rewrite /pgroup (card_isog isoGH). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
isog_pgroup
| |
isog_pcorepi : G \isog H -> 'O_pi(G) \isog 'O_pi(H).
Proof. exact: gFisog. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
isog_pcore
| |
isog_pseriespis : G \isog H -> pseries pis G \isog pseries pis H.
Proof. exact: gFisog. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
isog_pseries
| |
imprimitivity_systemQ :=
[&& partition Q S, [acts A, on Q | to^*] & 1 < #|Q| < #|S|].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
imprimitivity_system
| |
primitive:=
[transitive A, on S | to] && ~~ [exists Q, imprimitivity_system Q].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
primitive
| |
trans_prim_astabx :
x \in S -> [transitive G, on S | to] ->
[primitive G, on S | to] = maximal_eq 'C_G[x | to] G.
Proof.
move=> Sx trG; rewrite /primitive trG negb_exists.
apply/forallP/maximal_eqP=> /= [primG | [_ maxCx] Q].
split=> [|H sCH sHG]; first exact: subsetIl.
pose X := orbit to H x; pose Q := orbit (to^*)%act G X.
have Xx: x \in X by apply: orbit_refl.
have defH: 'N_(G)(X | to) = H.
have trH: [transitive H, on X | to] by apply/imsetP; exists x.
have sHN: H \subset 'N_G(X | to) by rewrite subsetI sHG atrans_acts.
move/(subgroup_transitiveP Xx sHN): (trH) => /= <-.
by rewrite mulSGid //= setIAC subIset ?sCH.
apply/imsetP; exists x => //; apply/eqP.
by rewrite eqEsubset imsetS // acts_sub_orbit ?subsetIr.
have [|/proper_card oCH] := eqVproper sCH; [by left | right].
apply/eqP; rewrite eqEcard sHG leqNgt.
apply: contra {primG}(primG Q) => oHG; apply/and3P; split; last first.
- rewrite card_orbit astab1_set defH -(@ltn_pmul2l #|H|) ?Lagrange // muln1.
rewrite oHG -(@ltn_pmul2l #|H|) ?Lagrange // -(card_orbit_stab to G x).
by rewrite -(atransP trG x Sx) mulnC card_orbit ltn_pmul2r.
- by apply/actsP=> a Ga Y; apply/orbit_transl/mem_orbit.
apply/and3P; split; last 1 first.
- rewrite orbit_sym; apply/imsetP=> [[a _]] /= defX.
by rewrite defX /setact imset0 inE in Xx.
- apply/eqP/setP=> y; apply/bigcupP/idP=> [[_ /imsetP[a Ga ->]] | Sy].
case/imsetP=> _ /imsetP[b Hb ->] ->.
by rewrite !(actsP (atrans_acts trG))
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
trans_prim_astab
| |
prim_trans_norm(H : {group aT}) :
[primitive G, on S | to] -> H <| G ->
H \subset 'C_G(S | to) \/ [transitive H, on S | to].
Proof.
move=> primG /andP[sHG nHG]; rewrite subsetI sHG.
have [trG _] := andP primG; have [x Sx defS] := imsetP trG.
move: primG; rewrite (trans_prim_astab Sx) // => /maximal_eqP[_].
case/(_ ('C_G[x | to] <*> H)%G) => /= [||cxH|]; first exact: joing_subl.
- by rewrite join_subG subsetIl.
- have{} cxH: H \subset 'C_G[x | to] by rewrite -cxH joing_subr.
rewrite subsetI sHG /= in cxH; left; apply/subsetP=> a Ha.
apply/astabP=> y Sy; have [b Gb ->] := atransP2 trG Sx Sy.
rewrite actCJV [to x (a ^ _)](astab1P _) ?(subsetP cxH) //.
by rewrite -mem_conjg (normsP nHG).
rewrite norm_joinEl 1?subIset ?nHG //.
by move/(subgroup_transitiveP Sx sHG trG); right.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
prim_trans_norm
| |
n_act(t : n.-tuple sT) a := [tuple of map (to^~ a) t].
Fact n_act_is_action : is_action setT n_act.
Proof.
by apply: is_total_action => [t|t a b]; apply: eq_from_tnth => i;
rewrite !tnth_map ?act1 ?actM.
Qed.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
n_act
| |
n_act_action:= Action n_act_is_action.
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
n_act_action
|
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