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 count... | 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 count... | 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/... | 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 count... | 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 count... | 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 count... | 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]> <... | 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 count... | 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 count... | 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 mov... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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.
b... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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) (car... | 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 count... | 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 count... | 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_abel... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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_dvd... | 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 count... | 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; ca... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 ->]; rewr... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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_in... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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/p... | 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 count... | 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 count... | 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 (subse... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 r... | 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 count... | 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 mu... | 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 count... | 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 ... | 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 count... | 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 count... | 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 count... | 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 count... | 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 r... | 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 count... | 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 count... | 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 count... | 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 count... | 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... | 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 count... | 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 joi... | 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 count... | 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 count... | 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 count... | 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 ... | 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 count... | 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 count... | 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 count... | 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 rewr... | 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 count... | 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; ... | 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 count... | 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... | 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 count... | 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 count... | 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 //= ce... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | 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 count... | solvable/abelian.v | isog_Ohm |
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