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 |
|---|---|---|---|---|---|---|
nElemSn G H : G \subset H -> 'E^n(G) \subset 'E^n(H).
Proof.
move=> sGH; apply/subsetP=> E /nElemP[p EpnG_E].
by apply/nElemP; exists p; rewrite // (subsetP (pnElemS _ _ sGH)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
nElemS
| |
nElemIn G H : 'E^n(G :&: H) = 'E^n(G) :&: subgroups H.
Proof.
apply/setP=> E; apply/nElemP/setIP=> [[p] | []].
by rewrite pnElemI; case/setIP; split=> //; apply/nElemP; exists p.
by case/nElemP=> p EpnG_E sHE; exists p; rewrite pnElemI inE EpnG_E.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
nElemI
| |
def_pnElemp n G : 'E_p^n(G) = 'E_p(G) :&: 'E^n(G).
Proof.
apply/setP=> E; rewrite inE in_setI; apply: andb_id2l => /pElemP[sEG abelE].
apply/idP/nElemP=> [|[q]]; first by exists p; rewrite !inE sEG abelE.
rewrite !inE -2!andbA => /and4P[_ /pgroupP qE _].
have [->|] := eqVneq E 1%G; first by rewrite cards1 !logn1.
case/(pgroup_pdiv (abelem_pgroup abelE)) => p_pr pE _.
by rewrite (eqnP (qE p p_pr pE)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
def_pnElem
| |
pmaxElemPp A E :
reflect (E \in 'E_p(A) /\ forall H, H \in 'E_p(A) -> E \subset H -> H :=: E)
(E \in 'E*_p(A)).
Proof. by rewrite [E \in 'E*_p(A)]inE; apply: (iffP maxgroupP). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pmaxElemP
| |
pmaxElem_existsp A D :
D \in 'E_p(A) -> {E | E \in 'E*_p(A) & D \subset E}.
Proof.
move=> EpD; have [E maxE sDE] := maxgroup_exists (EpD : mem 'E_p(A) D).
by exists E; rewrite // inE.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pmaxElem_exists
| |
pmaxElem_LdivPp G E :
prime p -> reflect ('Ldiv_p('C_G(E)) = E) (E \in 'E*_p(G)).
Proof.
move=> p_pr; apply: (iffP (pmaxElemP p G E)) => [[] | defE].
case/pElemP=> sEG abelE maxE; have [_ cEE eE] := and3P abelE.
apply/setP=> x; rewrite !inE -andbA; apply/and3P/idP=> [[Gx cEx xp] | Ex].
rewrite -(maxE (<[x]> <*> E)%G) ?joing_subr //.
by rewrite -cycle_subG joing_subl.
rewrite inE join_subG cycle_subG Gx sEG /=.
rewrite (cprod_abelem _ (cprodEY _)); last by rewrite centsC cycle_subG.
by rewrite cycle_abelem ?p_pr ?orbT // order_dvdn xp.
by rewrite (subsetP sEG) // (subsetP cEE) // (exponentP eE).
split=> [|H]; last first.
case/pElemP=> sHG /abelemP[// | cHH Hp1] sEH.
apply/eqP; rewrite eqEsubset sEH andbC /= -defE; apply/subsetP=> x Hx.
by rewrite 3!inE (subsetP sHG) // Hp1 ?(subsetP (centsS _ cHH)) /=.
apply/pElemP; split; first by rewrite -defE -setIA subsetIl.
apply/abelemP=> //; rewrite /abelian -{1 3}defE setIAC subsetIr.
by split=> //; apply/exponentP; rewrite -sub_LdivT setIAC subsetIr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pmaxElem_LdivP
| |
pmaxElemSp A B :
A \subset B -> 'E*_p(B) :&: subgroups A \subset 'E*_p(A).
Proof.
move=> sAB; apply/subsetP=> E /[!inE].
case/andP=> /maxgroupP[/pElemP[_ abelE] maxE] sEA.
apply/maxgroupP; rewrite inE sEA; split=> // D EpD.
by apply: maxE; apply: subsetP EpD; apply: pElemS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pmaxElemS
| |
pmaxElemJp A E x : ((E :^ x)%G \in 'E*_p(A :^ x)) = (E \in 'E*_p(A)).
Proof.
apply/pmaxElemP/pmaxElemP=> [] [EpE maxE].
rewrite pElemJ in EpE; split=> //= H EpH sEH; apply: (act_inj 'Js x).
by apply: maxE; rewrite ?conjSg ?pElemJ.
rewrite pElemJ; split=> // H; rewrite -(actKV 'JG x H) pElemJ conjSg => EpHx'.
by move/maxE=> /= ->.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pmaxElemJ
| |
grank_minB : 'm(<<B>>) <= #|B|.
Proof.
by rewrite /gen_rank; case: arg_minnP => [|_ _ -> //]; rewrite genGid.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
grank_min
| |
grank_witnessG : {B | <<B>> = G & #|B| = 'm(G)}.
Proof.
rewrite /gen_rank; case: arg_minnP => [|B defG _]; first by rewrite genGid.
by exists B; first apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
grank_witness
| |
p_rank_witnessp G : {E | E \in 'E_p^('r_p(G))(G)}.
Proof.
have [E EG_E mE]: {E | E \in 'E_p(G) & 'r_p(G) = logn p #|E| }.
by apply: eq_bigmax_cond; rewrite (cardD1 1%G) inE sub1G abelem1.
by exists E; rewrite inE EG_E -mE /=.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_witness
| |
p_rank_gePp n G : reflect (exists E, E \in 'E_p^n(G)) (n <= 'r_p(G)).
Proof.
apply: (iffP idP) => [|[E]]; last first.
by rewrite inE => /andP[Ep_E /eqP <-]; rewrite (bigmax_sup E).
have [D /pnElemP[sDG abelD <-]] := p_rank_witness p G.
by case/abelem_pnElem=> // E; exists E; apply: (subsetP (pnElemS _ _ sDG)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_geP
| |
p_rank_gt0p H : ('r_p(H) > 0) = (p \in \pi(H)).
Proof.
rewrite mem_primes cardG_gt0 /=; apply/p_rank_geP/andP=> [[E] | [p_pr]].
case/pnElemP=> sEG _; rewrite lognE; case: and3P => // [[-> _ pE] _].
by rewrite (dvdn_trans _ (cardSg sEG)).
case/Cauchy=> // x Hx ox; exists <[x]>%G; rewrite 2!inE [#|_|]ox cycle_subG.
by rewrite Hx (pfactorK 1) ?abelemE // cycle_abelian -ox exponent_dvdn.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_gt0
| |
p_rank1p : 'r_p([1 gT]) = 0.
Proof. by apply/eqP; rewrite eqn0Ngt p_rank_gt0 /= cards1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank1
| |
logn_le_p_rankp A E : E \in 'E_p(A) -> logn p #|E| <= 'r_p(A).
Proof. by move=> EpA_E; rewrite (bigmax_sup E). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
logn_le_p_rank
| |
p_rank_le_lognp G : 'r_p(G) <= logn p #|G|.
Proof.
have [E EpE] := p_rank_witness p G.
by have [sEG _ <-] := pnElemP EpE; apply: lognSg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_le_logn
| |
p_rank_abelemp G : p.-abelem G -> 'r_p(G) = logn p #|G|.
Proof.
move=> abelG; apply/eqP; rewrite eqn_leq andbC (bigmax_sup G)//.
by apply/bigmax_leqP=> E /[1!inE] /andP[/lognSg->].
by rewrite inE subxx.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_abelem
| |
p_rankSp A B : A \subset B -> 'r_p(A) <= 'r_p(B).
Proof.
move=> sAB; apply/bigmax_leqP=> E /(subsetP (pElemS p sAB)) EpB_E.
by rewrite (bigmax_sup E).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rankS
| |
p_rankElem_maxp A : 'E_p^('r_p(A))(A) \subset 'E*_p(A).
Proof.
apply/subsetP=> E /setIdP[EpE dimE].
apply/pmaxElemP; split=> // F EpF sEF; apply/eqP.
have pF: p.-group F by case/pElemP: EpF => _ /and3P[].
have pE: p.-group E by case/pElemP: EpE => _ /and3P[].
rewrite eq_sym eqEcard sEF dvdn_leq // (card_pgroup pE) (card_pgroup pF).
by rewrite (eqP dimE) dvdn_exp2l // logn_le_p_rank.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rankElem_max
| |
p_rankJp A x : 'r_p(A :^ x) = 'r_p(A).
Proof.
rewrite /p_rank (reindex_inj (act_inj 'JG x)).
by apply: eq_big => [E | E _]; rewrite ?cardJg ?pElemJ.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rankJ
| |
p_rank_Sylowp G H : p.-Sylow(G) H -> 'r_p(H) = 'r_p(G).
Proof.
move=> sylH; apply/eqP; rewrite eqn_leq (p_rankS _ (pHall_sub sylH)) /=.
apply/bigmax_leqP=> E /[1!inE] /andP[sEG abelE].
have [P sylP sEP] := Sylow_superset sEG (abelem_pgroup abelE).
have [x _ ->] := Sylow_trans sylP sylH.
by rewrite p_rankJ -(p_rank_abelem abelE) (p_rankS _ sEP).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_Sylow
| |
p_rank_Hallpi p G H : pi.-Hall(G) H -> p \in pi -> 'r_p(H) = 'r_p(G).
Proof.
move=> hallH pi_p; have [P sylP] := Sylow_exists p H.
by rewrite -(p_rank_Sylow sylP) (p_rank_Sylow (subHall_Sylow hallH pi_p sylP)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_Hall
| |
p_rank_pmaxElem_existsp r G :
'r_p(G) >= r -> exists2 E, E \in 'E*_p(G) & 'r_p(E) >= r.
Proof.
case/p_rank_geP=> D /setIdP[EpD /eqP <- {r}].
have [E EpE sDE] := pmaxElem_exists EpD; exists E => //.
case/pmaxElemP: EpE => /setIdP[_ abelE] _.
by rewrite (p_rank_abelem abelE) lognSg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_pmaxElem_exists
| |
rank1: 'r([1 gT]) = 0.
Proof. by rewrite ['r(1)]big1_seq // => p _; rewrite p_rank1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rank1
| |
p_rank_le_rankp G : 'r_p(G) <= 'r(G).
Proof.
case: (posnP 'r_p(G)) => [-> //|]; rewrite p_rank_gt0 mem_primes.
case/and3P=> p_pr _ pG; have lepg: p < #|G|.+1 by rewrite ltnS dvdn_leq.
by rewrite ['r(G)]big_mkord (bigmax_sup (Ordinal lepg)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_le_rank
| |
rank_gt0G : ('r(G) > 0) = (G :!=: 1).
Proof.
case: (eqsVneq G 1) => [-> |]; first by rewrite rank1.
case: (trivgVpdiv G) => [/eqP->// | [p p_pr]].
case/Cauchy=> // x Gx oxp _; apply: leq_trans (p_rank_le_rank p G).
have EpGx: <[x]>%G \in 'E_p(G).
by rewrite inE cycle_subG Gx abelemE // cycle_abelian -oxp exponent_dvdn.
by apply: leq_trans (logn_le_p_rank EpGx); rewrite -orderE oxp logn_prime ?eqxx.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rank_gt0
| |
rank_witnessG : {p | prime p & 'r(G) = 'r_p(G)}.
Proof.
have [p _ defmG]: {p : 'I_(#|G|.+1) | true & 'r(G) = 'r_p(G)}.
by rewrite ['r(G)]big_mkord; apply: eq_bigmax_cond; rewrite card_ord.
case: (eqsVneq G 1) => [-> | ]; first by exists 2; rewrite // rank1 p_rank1.
by rewrite -rank_gt0 defmG p_rank_gt0 mem_primes; case/andP; exists p.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rank_witness
| |
rank_pgroupp G : p.-group G -> 'r(G) = 'r_p(G).
Proof.
move=> pG; apply/eqP; rewrite eqn_leq p_rank_le_rank andbT.
rewrite ['r(G)]big_mkord; apply/bigmax_leqP=> [[q /= _] _].
case: (posnP 'r_q(G)) => [-> // |]; rewrite p_rank_gt0 mem_primes.
by case/and3P=> q_pr _ qG; rewrite (eqnP (pgroupP pG q q_pr qG)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rank_pgroup
| |
rank_Sylowp G P : p.-Sylow(G) P -> 'r(P) = 'r_p(G).
Proof.
move=> sylP; have pP := pHall_pgroup sylP.
by rewrite -(p_rank_Sylow sylP) -(rank_pgroup pP).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rank_Sylow
| |
rank_abelemp G : p.-abelem G -> 'r(G) = logn p #|G|.
Proof.
by move=> abelG; rewrite (rank_pgroup (abelem_pgroup abelG)) p_rank_abelem.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rank_abelem
| |
nt_pnElemp n E A : E \in 'E_p^n(A) -> n > 0 -> E :!=: 1.
Proof. by case/pnElemP=> _ /rank_abelem <- <-; rewrite rank_gt0. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
nt_pnElem
| |
rankJA x : 'r(A :^ x) = 'r(A).
Proof. by rewrite /rank cardJg; apply: eq_bigr => p _; rewrite p_rankJ. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rankJ
| |
rankSA B : A \subset B -> 'r(A) <= 'r(B).
Proof.
move=> sAB; rewrite /rank !big_mkord; apply/bigmax_leqP=> p _.
have leAB: #|A| < #|B|.+1 by rewrite ltnS subset_leq_card.
by rewrite (bigmax_sup (widen_ord leAB p)) ?p_rankS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rankS
| |
rank_gePn G : reflect (exists E, E \in 'E^n(G)) (n <= 'r(G)).
Proof.
apply: (iffP idP) => [|[E]].
have [p _ ->] := rank_witness G; case/p_rank_geP=> E.
by rewrite def_pnElem; case/setIP; exists E.
case/nElemP=> p /[1!inE] /andP[EpG_E /eqP <-].
by rewrite (leq_trans (logn_le_p_rank EpG_E)) ?p_rank_le_rank.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rank_geP
| |
exponent_morphimG : exponent (f @* G) %| exponent G.
Proof.
apply/exponentP=> _ /morphimP[x Dx Gx ->].
by rewrite -morphX // expg_exponent // morph1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_morphim
| |
morphim_LdivTn : f @* 'Ldiv_n() \subset 'Ldiv_n().
Proof.
apply/subsetP=> _ /morphimP[x Dx xn ->]; rewrite inE in xn.
by rewrite inE -morphX // (eqP xn) morph1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
morphim_LdivT
| |
morphim_Ldivn A : f @* 'Ldiv_n(A) \subset 'Ldiv_n(f @* A).
Proof.
by apply: subset_trans (morphimI f A _) (setIS _ _); apply: morphim_LdivT.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
morphim_Ldiv
| |
morphim_abelemp G : p.-abelem G -> p.-abelem (f @* G).
Proof.
case: (eqsVneq G 1) => [-> | ntG] abelG; first by rewrite morphim1 abelem1.
have [p_pr _ _] := pgroup_pdiv (abelem_pgroup abelG) ntG.
case/abelemP: abelG => // abG elemG; apply/abelemP; rewrite ?morphim_abelian //.
by split=> // _ /morphimP[x Dx Gx ->]; rewrite -morphX // elemG ?morph1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
morphim_abelem
| |
morphim_pElemp G E : E \in 'E_p(G) -> (f @* E)%G \in 'E_p(f @* G).
Proof.
by rewrite !inE => /andP[sEG abelE]; rewrite morphimS // morphim_abelem.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
morphim_pElem
| |
morphim_pnElemp n G E :
E \in 'E_p^n(G) -> {m | m <= n & (f @* E)%G \in 'E_p^m(f @* G)}.
Proof.
rewrite inE => /andP[EpE /eqP <-].
by exists (logn p #|f @* E|); rewrite ?logn_morphim // inE morphim_pElem /=.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
morphim_pnElem
| |
morphim_grankG : G \subset D -> 'm(f @* G) <= 'm(G).
Proof.
have [B defG <-] := grank_witness G; rewrite -defG gen_subG => sBD.
by rewrite morphim_gen ?morphimEsub ?(leq_trans (grank_min _)) ?leq_imset_card.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
morphim_grank
| |
exponent_injm: exponent (f @* G) = exponent G.
Proof. by apply/eqP; rewrite eqn_dvd -{3}defG !exponent_morphim. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_injm
| |
injm_Ldivn A : f @* 'Ldiv_n(A) = 'Ldiv_n(f @* A).
Proof.
apply/eqP; rewrite eqEsubset morphim_Ldiv.
rewrite -[f @* 'Ldiv_n(A)](morphpre_invm injf).
rewrite -sub_morphim_pre; last by rewrite subIset ?morphim_sub.
rewrite injmI ?injm_invm // setISS ?morphim_LdivT //.
by rewrite sub_morphim_pre ?morphim_sub // morphpre_invm.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_Ldiv
| |
injm_abelemp : p.-abelem (f @* G) = p.-abelem G.
Proof. by apply/idP/idP; first rewrite -{2}defG; apply: morphim_abelem. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_abelem
| |
injm_pElemp (E : {group aT}) :
E \subset D -> ((f @* E)%G \in 'E_p(f @* G)) = (E \in 'E_p(G)).
Proof.
move=> sED; apply/idP/idP=> EpE; last exact: morphim_pElem.
by rewrite -defG -(group_inj (morphim_invm injf sED)) morphim_pElem.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_pElem
| |
injm_pnElemp n (E : {group aT}) :
E \subset D -> ((f @* E)%G \in 'E_p^n(f @* G)) = (E \in 'E_p^n(G)).
Proof. by move=> sED; rewrite inE injm_pElem // card_injm ?inE. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_pnElem
| |
injm_nElemn (E : {group aT}) :
E \subset D -> ((f @* E)%G \in 'E^n(f @* G)) = (E \in 'E^n(G)).
Proof.
move=> sED; apply/nElemP/nElemP=> [] [p EpE];
by exists p; rewrite injm_pnElem in EpE *.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_nElem
| |
injm_pmaxElemp (E : {group aT}) :
E \subset D -> ((f @* E)%G \in 'E*_p(f @* G)) = (E \in 'E*_p(G)).
Proof.
move=> sED; have defE := morphim_invm injf sED.
apply/pmaxElemP/pmaxElemP=> [] [EpE maxE].
split=> [|H EpH sEH]; first by rewrite injm_pElem in EpE.
have sHD: H \subset D by apply: subset_trans (sGD); case/pElemP: EpH.
by rewrite -(morphim_invm injf sHD) [f @* H]maxE ?morphimS ?injm_pElem.
rewrite injm_pElem //; split=> // fH Ep_fH sfEH; have [sfHG _] := pElemP Ep_fH.
have sfHD : fH \subset f @* D by rewrite (subset_trans sfHG) ?morphimS.
rewrite -(morphpreK sfHD); congr (f @* _).
rewrite [_ @*^-1 fH]maxE -?sub_morphim_pre //.
by rewrite -injm_pElem ?subsetIl // (group_inj (morphpreK sfHD)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_pmaxElem
| |
injm_grank: 'm(f @* G) = 'm(G).
Proof. by apply/eqP; rewrite eqn_leq -{3}defG !morphim_grank ?morphimS. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_grank
| |
injm_p_rankp : 'r_p(f @* G) = 'r_p(G).
Proof.
apply/eqP; rewrite eqn_leq; apply/andP; split.
have [fE] := p_rank_witness p (f @* G); move: 'r_p(_) => n Ep_fE.
apply/p_rank_geP; exists (f @*^-1 fE)%G.
rewrite -injm_pnElem ?subsetIl ?(group_inj (morphpreK _)) //.
by case/pnElemP: Ep_fE => sfEG _ _; rewrite (subset_trans sfEG) ?morphimS.
have [E] := p_rank_witness p G; move: 'r_p(_) => n EpE.
apply/p_rank_geP; exists (f @* E)%G; rewrite injm_pnElem //.
by case/pnElemP: EpE => sEG _ _; rewrite (subset_trans sEG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_p_rank
| |
injm_rank: 'r(f @* G) = 'r(G).
Proof.
apply/eqP; rewrite eqn_leq; apply/andP; split.
by have [p _ ->] := rank_witness (f @* G); rewrite injm_p_rank p_rank_le_rank.
by have [p _ ->] := rank_witness G; rewrite -injm_p_rank p_rank_le_rank.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_rank
| |
exponent_isog: exponent G = exponent H.
Proof. by case/isogP: isoGH => f injf <-; rewrite exponent_injm. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_isog
| |
isog_abelemp : p.-abelem G = p.-abelem H.
Proof. by case/isogP: isoGH => f injf <-; rewrite injm_abelem. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
isog_abelem
| |
isog_grank: 'm(G) = 'm(H).
Proof. by case/isogP: isoGH => f injf <-; rewrite [RHS]injm_grank. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
isog_grank
| |
isog_p_rankp : 'r_p(G) = 'r_p(H).
Proof. by case/isogP: isoGH => f injf <-; rewrite injm_p_rank. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
isog_p_rank
| |
isog_rank: 'r(G) = 'r(H).
Proof. by case/isogP: isoGH => f injf <-; rewrite injm_rank. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
isog_rank
| |
exponent_quotientG H : exponent (G / H) %| exponent G.
Proof. exact: exponent_morphim. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_quotient
| |
quotient_LdivTn H : 'Ldiv_n() / H \subset 'Ldiv_n().
Proof. exact: morphim_LdivT. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
quotient_LdivT
| |
quotient_Ldivn A H : 'Ldiv_n(A) / H \subset 'Ldiv_n(A / H).
Proof. exact: morphim_Ldiv. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
quotient_Ldiv
| |
quotient_abelemG H : p.-abelem G -> p.-abelem (G / H).
Proof. exact: morphim_abelem. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
quotient_abelem
| |
quotient_pElemG H E : E \in 'E_p(G) -> (E / H)%G \in 'E_p(G / H).
Proof. exact: morphim_pElem. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
quotient_pElem
| |
logn_quotientG H : logn p #|G / H| <= logn p #|G|.
Proof. exact: logn_morphim. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
logn_quotient
| |
quotient_pnElemG H n E :
E \in 'E_p^n(G) -> {m | m <= n & (E / H)%G \in 'E_p^m(G / H)}.
Proof. exact: morphim_pnElem. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
quotient_pnElem
| |
quotient_grankG H : G \subset 'N(H) -> 'm(G / H) <= 'm(G).
Proof. exact: morphim_grank. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
quotient_grank
| |
p_rank_quotientG H : G \subset 'N(H) -> 'r_p(G) - 'r_p(H) <= 'r_p(G / H).
Proof.
move=> nHG; rewrite leq_subLR.
have [E EpE] := p_rank_witness p G; have{EpE} [sEG abelE <-] := pnElemP EpE.
rewrite -(LagrangeI E H) lognM ?cardG_gt0 //.
rewrite -card_quotient ?(subset_trans sEG) // leq_add ?logn_le_p_rank // !inE.
by rewrite subsetIr (abelemS (subsetIl E H)).
by rewrite quotientS ?quotient_abelem.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_quotient
| |
p_rank_dprodK H G : K \x H = G -> 'r_p(K) + 'r_p(H) = 'r_p(G).
Proof.
move=> defG; apply/eqP; rewrite eqn_leq -leq_subLR andbC.
have [_ defKH cKH tiKH] := dprodP defG; have nKH := cents_norm cKH.
rewrite {1}(isog_p_rank (quotient_isog nKH tiKH)) /= -quotientMidl defKH.
rewrite p_rank_quotient; last by rewrite -defKH mul_subG ?normG.
have [[E EpE] [F EpF]] := (p_rank_witness p K, p_rank_witness p H).
have [[sEK abelE <-] [sFH abelF <-]] := (pnElemP EpE, pnElemP EpF).
have defEF: E \x F = E <*> F.
by rewrite dprodEY ?(centSS sFH sEK) //; apply/trivgP; rewrite -tiKH setISS.
apply/p_rank_geP; exists (E <*> F)%G; rewrite !inE (dprod_abelem p defEF).
rewrite -lognM ?cargG_gt0 // (dprod_card defEF) abelE abelF eqxx.
by rewrite -(genGid G) -defKH genM_join genS ?setUSS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_dprod
| |
p_rank_p'quotientG H :
(p : nat)^'.-group H -> G \subset 'N(H) -> 'r_p(G / H) = 'r_p(G).
Proof.
move=> p'H nHG; have [P sylP] := Sylow_exists p G.
have [sPG pP _] := and3P sylP; have nHP := subset_trans sPG nHG.
have tiHP: H :&: P = 1 := coprime_TIg (p'nat_coprime p'H pP).
rewrite -(p_rank_Sylow sylP) -(p_rank_Sylow (quotient_pHall nHP sylP)).
by rewrite (isog_p_rank (quotient_isog nHP tiHP)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank_p'quotient
| |
Ohm_subG : 'Ohm_n(G) \subset G.
Proof. by rewrite gen_subG; apply/subsetP=> x /setIdP[]. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_sub
| |
Ohm1: 'Ohm_n([1 gT]) = 1. Proof. exact: (trivgP (Ohm_sub _)). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm1
| |
Ohm_idG : 'Ohm_n('Ohm_n(G)) = 'Ohm_n(G).
Proof.
apply/eqP; rewrite eqEsubset Ohm_sub genS //.
by apply/subsetP=> x /setIdP[Gx oxn]; rewrite inE mem_gen // inE Gx.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_id
| |
Ohm_contrT G (f : {morphism G >-> rT}) :
f @* 'Ohm_n(G) \subset 'Ohm_n(f @* G).
Proof.
rewrite morphim_gen ?genS //; last by rewrite -gen_subG Ohm_sub.
apply/subsetP=> fx /morphimP[x Gx]; rewrite inE Gx /=.
case/OhmPredP=> p p_pr xpn_1 -> {fx}.
rewrite inE morphimEdom imset_f //=; apply/OhmPredP; exists p => //.
by rewrite -morphX // xpn_1 morph1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_cont
| |
OhmSH G : H \subset G -> 'Ohm_n(H) \subset 'Ohm_n(G).
Proof.
move=> sHG; apply: genS; apply/subsetP=> x /[!inE] /andP[Hx ->].
by rewrite (subsetP sHG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
OhmS
| |
OhmEp G : p.-group G -> 'Ohm_n(G) = <<'Ldiv_(p ^ n)(G)>>.
Proof.
move=> pG; congr <<_>>; apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
have [-> | ntx] := eqVneq x 1; first by rewrite !expg1n.
by rewrite (pdiv_p_elt (mem_p_elt pG Gx)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
OhmE
| |
OhmEabelianp G :
p.-group G -> abelian 'Ohm_n(G) -> 'Ohm_n(G) = 'Ldiv_(p ^ n)(G).
Proof.
move=> pG; rewrite (OhmE pG) abelian_gen => cGGn; rewrite gen_set_id //.
rewrite -(setIidPr (subset_gen 'Ldiv_(p ^ n)(G))) setIA.
by rewrite [_ :&: G](setIidPl _) ?gen_subG ?subsetIl // group_Ldiv ?abelian_gen.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
OhmEabelian
| |
Ohm_p_cyclep x :
p.-elt x -> 'Ohm_n(<[x]>) = <[x ^+ (p ^ (logn p #[x] - n))]>.
Proof.
move=> p_x; apply/eqP; rewrite (OhmE p_x) eqEsubset cycle_subG mem_gen.
rewrite gen_subG andbT; apply/subsetP=> y /LdivP[x_y ypn].
case: (leqP (logn p #[x]) n) => [|lt_n_x].
by rewrite -subn_eq0 => /eqP->.
have p_pr: prime p by move: lt_n_x; rewrite lognE; case: (prime p).
have def_y: <[y]> = <[x ^+ (#[x] %/ #[y])]>.
apply: congr_group; apply/set1P.
by rewrite -cycle_sub_group ?cardSg ?inE ?cycle_subG ?x_y /=.
rewrite -cycle_subG def_y cycle_subG -{1}(part_pnat_id p_x) p_part.
rewrite -{1}(subnK (ltnW lt_n_x)) expnD -muln_divA ?order_dvdn ?ypn //.
by rewrite expgM mem_cycle.
rewrite !inE mem_cycle -expgM -expnD addnC -maxnE -order_dvdn.
by rewrite -{1}(part_pnat_id p_x) p_part dvdn_exp2l ?leq_maxr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_p_cycle
| |
Ohm_dprodA B G : A \x B = G -> 'Ohm_n(A) \x 'Ohm_n(B) = 'Ohm_n(G).
Proof.
case/dprodP => [[H K -> ->{A B}]] <- cHK tiHK.
rewrite dprodEY //; last first.
- by apply/trivgP; rewrite -tiHK setISS ?Ohm_sub.
- by rewrite (subset_trans (subset_trans _ cHK)) ?centS ?Ohm_sub.
apply/eqP; rewrite -(cent_joinEr cHK) eqEsubset join_subG /=.
rewrite !OhmS ?joing_subl ?joing_subr //= cent_joinEr //= -genM_join genS //.
apply/subsetP=> _ /setIdP[/imset2P[x y Hx Ky ->] /OhmPredP[p p_pr /eqP]].
have cxy: commute x y by red; rewrite -(centsP cHK).
rewrite ?expgMn // -eq_invg_mul => /eqP def_x.
have ypn1: y ^+ (p ^ n) = 1.
by apply/set1P; rewrite -[[set 1]]tiHK inE -{1}def_x groupV !groupX.
have xpn1: x ^+ (p ^ n) = 1 by rewrite -[x ^+ _]invgK def_x ypn1 invg1.
by rewrite mem_mulg ?mem_gen // inE (Hx, Ky); apply/OhmPredP; exists p.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_dprod
| |
Mho_subG : 'Mho^n(G) \subset G.
Proof.
rewrite gen_subG; apply/subsetP=> _ /imsetP[x /setIdP[Gx _] ->].
exact: groupX.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_sub
| |
Mho1: 'Mho^n([1 gT]) = 1. Proof. exact: (trivgP (Mho_sub _)). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho1
| |
morphim_MhorT D G (f : {morphism D >-> rT}) :
G \subset D -> f @* 'Mho^n(G) = 'Mho^n(f @* G).
Proof.
move=> sGD; have sGnD := subset_trans (Mho_sub G) sGD.
apply/eqP; rewrite eqEsubset {1}morphim_gen -1?gen_subG // !gen_subG.
apply/andP; split; apply/subsetP=> y.
case/morphimP=> xpn _ /imsetP[x /setIdP[Gx]].
set p := pdiv _ => p_x -> -> {xpn y}; have Dx := subsetP sGD x Gx.
by rewrite morphX // Mho_p_elt ?morph_p_elt ?mem_morphim.
case/imsetP=> _ /setIdP[/morphimP[x Dx Gx ->]].
set p := pdiv _ => p_fx ->{y}; rewrite -(constt_p_elt p_fx) -morph_constt //.
by rewrite -morphX ?mem_morphim ?Mho_p_elt ?groupX ?p_elt_constt.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
morphim_Mho
| |
Mho_contrT G (f : {morphism G >-> rT}) :
f @* 'Mho^n(G) \subset 'Mho^n(f @* G).
Proof. by rewrite morphim_Mho. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_cont
| |
MhoSH G : H \subset G -> 'Mho^n(H) \subset 'Mho^n(G).
Proof.
move=> sHG; apply: genS; apply: imsetS; apply/subsetP=> x.
by rewrite !inE => /andP[Hx]; rewrite (subsetP sHG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
MhoS
| |
MhoEp G : p.-group G -> 'Mho^n(G) = <<[set x ^+ (p ^ n) | x in G]>>.
Proof.
move=> pG; apply/eqP; rewrite eqEsubset !gen_subG; apply/andP.
do [split; apply/subsetP=> xpn; case/imsetP=> x] => [|Gx ->]; last first.
by rewrite Mho_p_elt ?(mem_p_elt pG).
case/setIdP=> Gx _ ->; have [-> | ntx] := eqVneq x 1; first by rewrite expg1n.
by rewrite (pdiv_p_elt (mem_p_elt pG Gx) ntx) mem_gen //; apply: imset_f.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
MhoE
| |
MhoEabelianp G :
p.-group G -> abelian G -> 'Mho^n(G) = [set x ^+ (p ^ n) | x in G].
Proof.
move=> pG cGG; rewrite (MhoE pG); rewrite gen_set_id //; apply/group_setP.
split=> [|xn yn]; first by apply/imsetP; exists 1; rewrite ?expg1n.
case/imsetP=> x Gx ->; case/imsetP=> y Gy ->.
by rewrite -expgMn; [apply: imset_f; rewrite groupM | apply: (centsP cGG)].
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
MhoEabelian
| |
trivg_MhoG : 'Mho^n(G) == 1 -> 'Ohm_n(G) == G.
Proof.
rewrite -subG1 gen_subG eqEsubset Ohm_sub /= => Gp1.
rewrite -{1}(Sylow_gen G) genS //; apply/bigcupsP=> P.
case/SylowP=> p p_pr /and3P[sPG pP _]; apply/subsetP=> x Px.
have Gx := subsetP sPG x Px; rewrite inE Gx //=.
rewrite (sameP eqP set1P) (subsetP Gp1) ?mem_gen //; apply: imset_f.
by rewrite inE Gx; apply: pgroup_p (mem_p_elt pP Px).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
trivg_Mho
| |
Mho_p_cyclep x : p.-elt x -> 'Mho^n(<[x]>) = <[x ^+ (p ^ n)]>.
Proof.
move=> p_x.
apply/eqP; rewrite (MhoE p_x) eqEsubset cycle_subG mem_gen; last first.
by apply: imset_f; apply: cycle_id.
rewrite gen_subG andbT; apply/subsetP=> _ /imsetP[_ /cycleP[k ->] ->].
by rewrite -expgM mulnC expgM mem_cycle.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_p_cycle
| |
Mho_cprodA B G : A \* B = G -> 'Mho^n(A) \* 'Mho^n(B) = 'Mho^n(G).
Proof.
case/cprodP => [[H K -> ->{A B}]] <- cHK; rewrite cprodEY //; last first.
by rewrite (subset_trans (subset_trans _ cHK)) ?centS ?Mho_sub.
apply/eqP; rewrite -(cent_joinEr cHK) eqEsubset join_subG /=.
rewrite !MhoS ?joing_subl ?joing_subr //= cent_joinEr // -genM_join.
apply: genS; apply/subsetP=> xypn /imsetP[_ /setIdP[/imset2P[x y Hx Ky ->]]].
move/constt_p_elt; move: (pdiv _) => p <- ->.
have cxy: commute x y by red; rewrite -(centsP cHK).
rewrite consttM // expgMn; last exact: commuteX2.
by rewrite mem_mulg ?Mho_p_elt ?groupX ?p_elt_constt.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_cprod
| |
Mho_dprodA B G : A \x B = G -> 'Mho^n(A) \x 'Mho^n(B) = 'Mho^n(G).
Proof.
case/dprodP => [[H K -> ->{A B}]] defG cHK tiHK.
rewrite dprodEcp; first by apply: Mho_cprod; rewrite cprodE.
by apply/trivgP; rewrite -tiHK setISS ?Mho_sub.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_dprod
| |
Ohm_igFuni := [igFun by Ohm_sub i & Ohm_cont i].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_igFun
| |
Ohm_gFuni := [gFun by Ohm_cont i].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_gFun
| |
Ohm_mgFuni := [mgFun by OhmS i].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_mgFun
| |
Mho_igFuni := [igFun by Mho_sub i & Mho_cont i].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_igFun
| |
Mho_gFuni := [gFun by Mho_cont i].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_gFun
| |
Mho_mgFuni := [mgFun by MhoS i].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_mgFun
| |
Ohm_char: 'Ohm_n(G) \char G. Proof. exact: gFchar. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_char
| |
Ohm_normal: 'Ohm_n(G) <| G. Proof. exact: gFnormal. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_normal
| |
Mho_char: 'Mho^n(G) \char G. Proof. exact: gFchar. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_char
| |
Mho_normal: 'Mho^n(G) <| G. Proof. exact: gFnormal. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_normal
| |
morphim_Ohm(f : {morphism D >-> rT}) :
G \subset D -> f @* 'Ohm_n(G) \subset 'Ohm_n(f @* G).
Proof. exact: morphimF. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
morphim_Ohm
| |
injm_Ohm(f : {morphism D >-> rT}) :
'injm f -> G \subset D -> f @* 'Ohm_n(G) = 'Ohm_n(f @* G).
Proof. by move=> injf; apply: injmF. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
injm_Ohm
| |
isog_Ohm(H : {group rT}) : G \isog H -> 'Ohm_n(G) \isog 'Ohm_n(H).
Proof. exact: gFisog. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
isog_Ohm
|
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