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
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|---|---|---|---|---|---|---|
normT: 'N([set: gT]) = [set: gT].
Proof. by apply/eqP; rewrite -subTset normG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normT
| |
normsGA G : A \subset G -> A \subset 'N(G).
Proof. by move=> sAG; apply: subset_trans (normG G). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsG
| |
normCA B : A \subset 'N(B) -> commute A B.
Proof.
move/subsetP=> nBA; apply/setP=> u.
apply/mulsgP/mulsgP=> [[x y Ax By] | [y x By Ax]] -> {u}.
by exists (y ^ x^-1) x; rewrite -?conjgCV // memJ_norm // groupV nBA.
by exists x (y ^ x); rewrite -?conjgC // memJ_norm // nBA.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normC
| |
norm_joinElG H : G \subset 'N(H) -> G <*> H = G * H.
Proof. by move/normC/comm_joingE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norm_joinEl
| |
norm_joinErG H : H \subset 'N(G) -> G <*> H = G * H.
Proof. by move/normC=> cHG; apply: comm_joingE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norm_joinEr
| |
norm_rlcosetG x : x \in 'N(G) -> G :* x = x *: G.
Proof. by rewrite -sub1set => /normC. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norm_rlcoset
| |
rcoset_mulG x y : x \in 'N(G) -> (G :* x) * (G :* y) = G :* (x * y).
Proof.
move/norm_rlcoset=> GxxG.
by rewrite mulgA -(mulgA _ _ G) -GxxG mulgA mulGid -mulgA mulg_set1.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
rcoset_mul
| |
normJA x : 'N(A :^ x) = 'N(A) :^ x.
Proof.
by apply/setP=> y; rewrite mem_conjg !inE -conjsgM conjgCV conjsgM conjSg.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normJ
| |
norm_conj_normx A B :
x \in 'N(A) -> (A \subset 'N(B :^ x)) = (A \subset 'N(B)).
Proof. by move=> Nx; rewrite normJ -sub_conjgV (normP _) ?groupV. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norm_conj_norm
| |
norm_genA : 'N(A) \subset 'N(<<A>>).
Proof. by apply/normsP=> x Nx; rewrite -genJ (normP Nx). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norm_gen
| |
class_normx G : G \subset 'N(x ^: G).
Proof. by apply/normsP=> y; apply: classGidr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
class_norm
| |
class_normalx G : x \in G -> x ^: G <| G.
Proof. by move=> Gx; rewrite /normal class_norm class_subG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
class_normal
| |
class_sub_normG A x : G \subset 'N(A) -> (x ^: G \subset A) = (x \in A).
Proof.
move=> nAG; apply/subsetP/idP=> [-> // | Ax xy]; first exact: class_refl.
by case/imsetP=> y Gy ->; rewrite memJ_norm ?(subsetP nAG).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
class_sub_norm
| |
class_support_normA G : G \subset 'N(class_support A G).
Proof. by apply/normsP; apply: class_supportGidr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
class_support_norm
| |
class_support_sub_normA B G :
A \subset G -> B \subset 'N(G) -> class_support A B \subset G.
Proof.
move=> sAG nGB; rewrite class_supportEr.
by apply/bigcupsP=> x Bx; rewrite -(normsP nGB x Bx) conjSg.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
class_support_sub_norm
| |
norms_gen: A \subset 'N(<<B>>).
Proof. exact: subset_trans nBA (norm_gen B). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norms_gen
| |
norms_norm: A \subset 'N('N(B)).
Proof. by apply/normsP=> x Ax; rewrite -normJ (normsP nBA). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norms_norm
| |
normsI: A \subset 'N(B :&: C).
Proof. by apply/normsP=> x Ax; rewrite conjIg !(normsP _ x Ax). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsI
| |
normsU: A \subset 'N(B :|: C).
Proof. by apply/normsP=> x Ax; rewrite conjUg !(normsP _ x Ax). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsU
| |
normsIs: B \subset 'N(D) -> A :&: B \subset 'N(C :&: D).
Proof.
move/normsP=> nDB; apply/normsP=> x; case/setIP=> Ax Bx.
by rewrite conjIg (normsP nCA) ?nDB.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsIs
| |
normsD: A \subset 'N(B :\: C).
Proof. by apply/normsP=> x Ax; rewrite conjDg !(normsP _ x Ax). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsD
| |
normsM: A \subset 'N(B * C).
Proof. by apply/normsP=> x Ax; rewrite conjsMg !(normsP _ x Ax). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsM
| |
normsY: A \subset 'N(B <*> C).
Proof. by apply/normsP=> x Ax; rewrite -genJ conjUg !(normsP _ x Ax). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsY
| |
normsR: A \subset 'N([~: B, C]).
Proof. by apply/normsP=> x Ax; rewrite conjsRg !(normsP _ x Ax). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsR
| |
norms_class_support: A \subset 'N(class_support B C).
Proof.
apply/subsetP=> x Ax; rewrite inE sub_conjg class_supportEr.
apply/bigcupsP=> y Cy; rewrite -sub_conjg -conjsgM conjgC conjsgM.
by rewrite (normsP nBA) // bigcup_sup ?memJ_norm ?(subsetP nCA).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norms_class_support
| |
normsIGA B G : A \subset 'N(B) -> A :&: G \subset 'N(B :&: G).
Proof. by move/normsIs->; rewrite ?normG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsIG
| |
normsGIA B G : A \subset 'N(B) -> G :&: A \subset 'N(G :&: B).
Proof. by move=> nBA; rewrite !(setIC G) normsIG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsGI
| |
norms_bigcapI r (P : pred I) A (B_ : I -> {set gT}) :
A \subset \bigcap_(i <- r | P i) 'N(B_ i) ->
A \subset 'N(\bigcap_(i <- r | P i) B_ i).
Proof.
elim/big_rec2: _ => [|i B N _ IH /subsetIP[nBiA /IH]]; last exact: normsI.
by rewrite normT.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norms_bigcap
| |
norms_bigcupI r (P : pred I) A (B_ : I -> {set gT}) :
A \subset \bigcap_(i <- r | P i) 'N(B_ i) ->
A \subset 'N(\bigcup_(i <- r | P i) B_ i).
Proof.
move=> nBA; rewrite -normCs setC_bigcup norms_bigcap //.
by rewrite (eq_bigr _ (fun _ _ => normCs _)).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norms_bigcup
| |
normsD1A B : A \subset 'N(B) -> A \subset 'N(B^#).
Proof. by move/normsD->; rewrite ?norms1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsD1
| |
normD1A : 'N(A^#) = 'N(A).
Proof.
apply/eqP; rewrite eqEsubset normsD1 //.
rewrite -{2}(setID A 1) setIC normsU //; apply/normsP=> x _; apply/setP=> y.
by rewrite conjIg conjs1g !inE mem_conjg; case: eqP => // ->; rewrite conj1g.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normD1
| |
normalPA B : reflect (A \subset B /\ {in B, normalised A}) (A <| B).
Proof. by apply: (iffP andP)=> [] [sAB]; move/normsP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalP
| |
normal_subA B : A <| B -> A \subset B.
Proof. by case/andP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normal_sub
| |
normal_normA B : A <| B -> B \subset 'N(A).
Proof. by case/andP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normal_norm
| |
normalSG H K : K \subset H -> H \subset G -> K <| G -> K <| H.
Proof.
by move=> sKH sHG /andP[_ nKG]; rewrite /(K <| _) sKH (subset_trans sHG).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalS
| |
normal1G : 1 <| G.
Proof. by rewrite /normal sub1set group1 norms1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normal1
| |
normal_reflG : G <| G.
Proof. by rewrite /(G <| _) normG subxx. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normal_refl
| |
normalGG : G <| 'N(G).
Proof. by rewrite /(G <| _) normG subxx. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalG
| |
normalSGG H : H \subset G -> H <| 'N_G(H).
Proof. by move=> sHG; rewrite /normal subsetI sHG normG subsetIr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalSG
| |
normalJA B x : (A :^ x <| B :^ x) = (A <| B).
Proof. by rewrite /normal normJ !conjSg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalJ
| |
normalMG A B : A <| G -> B <| G -> A * B <| G.
Proof.
by case/andP=> sAG nAG /andP[sBG nBG]; rewrite /normal mul_subG ?normsM.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalM
| |
normalYG A B : A <| G -> B <| G -> A <*> B <| G.
Proof.
by case/andP=> sAG ? /andP[sBG ?]; rewrite /normal join_subG sAG sBG ?normsY.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalY
| |
normalYlG H : (H <| H <*> G) = (G \subset 'N(H)).
Proof. by rewrite /normal joing_subl join_subG normG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalYl
| |
normalYrG H : (H <| G <*> H) = (G \subset 'N(H)).
Proof. by rewrite joingC normalYl. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalYr
| |
normalIG A B : A <| G -> B <| G -> A :&: B <| G.
Proof.
by case/andP=> sAG nAG /andP[_ nBG]; rewrite /normal subIset ?sAG // normsI.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalI
| |
norm_normalIG A : G \subset 'N(A) -> G :&: A <| G.
Proof. by move=> nAG; rewrite /normal subsetIl normsI ?normG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norm_normalI
| |
normalGIG H A : H \subset G -> A <| G -> H :&: A <| H.
Proof.
by move=> sHG /andP[_ nAG]; apply: norm_normalI (subset_trans sHG nAG).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalGI
| |
normal_subnormG H : (H <| 'N_G(H)) = (H \subset G).
Proof. by rewrite /normal subsetIr subsetI normG !andbT. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normal_subnorm
| |
normalD1A G : (A^# <| G) = (A <| G).
Proof. by rewrite /normal normD1 subDset (setUidPr (sub1G G)). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normalD1
| |
gcore_subA G : gcore A G \subset A.
Proof. by rewrite (bigcap_min 1) ?conjsg1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
gcore_sub
| |
gcore_normA G : G \subset 'N(gcore A G).
Proof.
apply/subsetP=> x Gx; rewrite inE; apply/bigcapsP=> y Gy.
by rewrite sub_conjg -conjsgM bigcap_inf ?groupM ?groupV.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
gcore_norm
| |
gcore_normalA G : A \subset G -> gcore A G <| G.
Proof.
by move=> sAG; rewrite /normal gcore_norm (subset_trans (gcore_sub A G)).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
gcore_normal
| |
gcore_maxA B G : B \subset A -> G \subset 'N(B) -> B \subset gcore A G.
Proof.
move=> sBA nBG; apply/bigcapsP=> y Gy.
by rewrite -sub_conjgV (normsP nBG) ?groupV.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
gcore_max
| |
sub_gcoreA B G :
G \subset 'N(B) -> (B \subset gcore A G) = (B \subset A).
Proof.
move=> nBG; apply/idP/idP=> [sBAG | sBA]; last exact: gcore_max.
exact: subset_trans (gcore_sub A G).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
sub_gcore
| |
rcoset_index2G H x :
H \subset G -> #|G : H| = 2 -> x \in G :\: H -> H :* x = G :\: H.
Proof.
move=> sHG indexHG => /setDP[Gx notHx]; apply/eqP.
rewrite eqEcard -(leq_add2l #|G :&: H|) cardsID -(LagrangeI G H) indexHG muln2.
rewrite (setIidPr sHG) card_rcoset addnn leqnn andbT.
apply/subsetP=> _ /rcosetP[y Hy ->]; apply/setDP.
by rewrite !groupMl // (subsetP sHG).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
rcoset_index2
| |
index2_normalG H : H \subset G -> #|G : H| = 2 -> H <| G.
Proof.
move=> sHG indexHG; rewrite /normal sHG; apply/subsetP=> x Gx.
case Hx: (x \in H); first by rewrite inE conjGid.
rewrite inE conjsgE mulgA -sub_rcosetV -invg_rcoset.
by rewrite !(rcoset_index2 sHG) ?inE ?groupV ?Hx // invDg !invGid.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
index2_normal
| |
cent1Px y : reflect (commute x y) (x \in 'C[y]).
Proof.
rewrite [x \in _]inE conjg_set1 sub1set !inE (sameP eqP conjg_fixP)commg1_sym.
exact: commgP.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent1P
| |
cent1idx : x \in 'C[x]. Proof. exact/cent1P. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent1id
| |
cent1Ex y : (x \in 'C[y]) = (x * y == y * x).
Proof. by rewrite (sameP (cent1P x y) eqP). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent1E
| |
cent1Cx y : (x \in 'C[y]) = (y \in 'C[x]).
Proof. by rewrite !cent1E eq_sym. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent1C
| |
centraliser_groupA : {group _} := Eval hnf in [group of 'C(A)].
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centraliser_group
| |
cent_set1x : 'C([set x]) = 'C[x].
Proof. by apply: big_pred1 => y /=; rewrite !inE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent_set1
| |
cent1Jx y : 'C[x ^ y] = 'C[x] :^ y.
Proof. by rewrite -conjg_set1 normJ. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent1J
| |
centPA x : reflect (centralises x A) (x \in 'C(A)).
Proof. by apply: (iffP bigcapP) => cxA y /cxA/cent1P. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centP
| |
centsPA B : reflect {in A, centralised B} (A \subset 'C(B)).
Proof. by apply: (iffP subsetP) => cAB x /cAB/centP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centsP
| |
centsCA B : (A \subset 'C(B)) = (B \subset 'C(A)).
Proof. by apply/centsP/centsP=> cAB x ? y ?; rewrite /commute -cAB. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centsC
| |
cents1A : A \subset 'C(1).
Proof. by rewrite centsC sub1G. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cents1
| |
cent1T: 'C(1) = setT :> {set gT}.
Proof. by apply/eqP; rewrite -subTset cents1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent1T
| |
cent11T: 'C[1] = setT :> {set gT}.
Proof. by rewrite -cent_set1 cent1T. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent11T
| |
cent_subA : 'C(A) \subset 'N(A).
Proof.
apply/subsetP=> x /centP cAx; rewrite inE.
by apply/subsetP=> _ /imsetP[y Ay ->]; rewrite /conjg -cAx ?mulKg.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent_sub
| |
cents_normA B : A \subset 'C(B) -> A \subset 'N(B).
Proof. by move=> cAB; apply: subset_trans (cent_sub B). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cents_norm
| |
centCA B : A \subset 'C(B) -> commute A B.
Proof. by move=> cAB; apply: normC (cents_norm cAB). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centC
| |
cent_joinElG H : G \subset 'C(H) -> G <*> H = G * H.
Proof. by move=> cGH; apply: norm_joinEl (cents_norm cGH). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent_joinEl
| |
cent_joinErG H : H \subset 'C(G) -> G <*> H = G * H.
Proof. by move=> cGH; apply: norm_joinEr (cents_norm cGH). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent_joinEr
| |
centJA x : 'C(A :^ x) = 'C(A) :^ x.
Proof.
apply/setP=> y; rewrite mem_conjg; apply/centP/centP=> cAy z Az.
apply: (conjg_inj x).
by rewrite conjMg [in RHS]conjMg conjgKV cAy ?memJ_conjg.
by apply: (conjg_inj x^-1); rewrite 2!conjMg cAy -?mem_conjg.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centJ
| |
cent_normA : 'N(A) \subset 'N('C(A)).
Proof. by apply/normsP=> x nCx; rewrite -centJ (normP nCx). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent_norm
| |
norms_centA B : A \subset 'N(B) -> A \subset 'N('C(B)).
Proof. by move=> nBA; apply: subset_trans nBA (cent_norm B). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norms_cent
| |
cent_normalA : 'C(A) <| 'N(A).
Proof. by rewrite /(_ <| _) cent_sub cent_norm. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent_normal
| |
centSA B : B \subset A -> 'C(A) \subset 'C(B).
Proof. by move=> sAB; rewrite centsC (subset_trans sAB) 1?centsC. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centS
| |
centsSA B C : A \subset B -> C \subset 'C(B) -> C \subset 'C(A).
Proof. by move=> sAB cCB; apply: subset_trans cCB (centS sAB). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centsS
| |
centSSA B C D :
A \subset C -> B \subset D -> C \subset 'C(D) -> A \subset 'C(B).
Proof. by move=> sAC sBD cCD; apply: subset_trans (centsS sBD cCD). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centSS
| |
centIA B : 'C(A) <*> 'C(B) \subset 'C(A :&: B).
Proof. by rewrite gen_subG subUset !centS ?(subsetIl, subsetIr). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centI
| |
centUA B : 'C(A :|: B) = 'C(A) :&: 'C(B).
Proof.
apply/eqP; rewrite eqEsubset subsetI 2?centS ?(subsetUl, subsetUr) //=.
by rewrite centsC subUset -centsC subsetIl -centsC subsetIr.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centU
| |
cent_genA : 'C(<<A>>) = 'C(A).
Proof. by apply/setP=> x; rewrite -!sub1set centsC gen_subG centsC. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent_gen
| |
cent_cyclex : 'C(<[x]>) = 'C[x].
Proof. by rewrite cent_gen cent_set1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent_cycle
| |
sub_cent1A x : (A \subset 'C[x]) = (x \in 'C(A)).
Proof. by rewrite -cent_cycle centsC cycle_subG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
sub_cent1
| |
cents_cyclex y : commute x y -> <[x]> \subset 'C(<[y]>).
Proof. by move=> cxy; rewrite cent_cycle cycle_subG; apply/cent1P. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cents_cycle
| |
cycle_abelianx : abelian <[x]>.
Proof. exact: cents_cycle. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycle_abelian
| |
centYA B : 'C(A <*> B) = 'C(A) :&: 'C(B).
Proof. by rewrite cent_gen centU. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centY
| |
centMG H : 'C(G * H) = 'C(G) :&: 'C(H).
Proof. by rewrite -cent_gen genM_join centY. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
centM
| |
cent_classPx G : reflect (x ^: G = [set x]) (x \in 'C(G)).
Proof.
apply: (iffP (centP _ _)) => [Cx | Cx1 y Gy].
apply/eqP; rewrite eqEsubset sub1set class_refl andbT.
by apply/subsetP=> _ /imsetP[y Gy ->]; rewrite !inE conjgE Cx ?mulKg.
by apply/commgP/conjg_fixP/set1P; rewrite -Cx1; apply/imsetP; exists y.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cent_classP
| |
commG1PA B : reflect ([~: A, B] = 1) (A \subset 'C(B)).
Proof.
apply: (iffP (centsP A B)) => [cAB | cAB1 x Ax y By].
apply/trivgP; rewrite gen_subG; apply/subsetP=> _ /imset2P[x y Ax Ay ->].
by rewrite inE; apply/commgP; apply: cAB.
by apply/commgP; rewrite -in_set1 -[[set 1]]cAB1 mem_commg.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
commG1P
| |
abelianEA : abelian A = (A \subset 'C(A)). Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
abelianE
| |
abelian1: abelian [1 gT]. Proof. exact: sub1G. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
abelian1
| |
abelianSA B : A \subset B -> abelian B -> abelian A.
Proof. by move=> sAB; apply: centSS. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
abelianS
| |
abelianJA x : abelian (A :^ x) = abelian A.
Proof. by rewrite /abelian centJ conjSg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
abelianJ
| |
abelian_genA : abelian <<A>> = abelian A.
Proof. by rewrite /abelian cent_gen gen_subG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
abelian_gen
| |
abelianYA B :
abelian (A <*> B) = [&& abelian A, abelian B & B \subset 'C(A)].
Proof.
rewrite /abelian join_subG /= centY !subsetI -!andbA; congr (_ && _).
by rewrite centsC andbA andbb andbC.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
abelianY
| |
abelianMG H :
abelian (G * H) = [&& abelian G, abelian H & H \subset 'C(G)].
Proof. by rewrite -abelian_gen genM_join abelianY. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
abelianM
| |
sub_abelian_cent: C \subset A -> A \subset 'C(C).
Proof. by move=> sCA; rewrite centsC (subset_trans sCA). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
sub_abelian_cent
|
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