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 |
|---|---|---|---|---|---|---|
quotientIGA G : H \subset G -> (A :&: G) / H = A / H :&: G / H.
Proof. by rewrite -{1}ker_coset; apply: morphimIG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientIG
| |
quotientDA B : A / H :\: B / H \subset (A :\: B) / H.
Proof. exact: morphimD. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientD
| |
quotientD1A : (A / H)^# \subset A^# / H.
Proof. exact: morphimD1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientD1
| |
quotientDGA G : H \subset G -> (A :\: G) / H = A / H :\: G / H.
Proof. by rewrite -{1}ker_coset; apply: morphimDG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientDG
| |
quotientKA : A \subset 'N(H) -> coset H @*^-1 (A / H) = H * A.
Proof. by rewrite -{8}ker_coset; apply: morphimK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientK
| |
quotientYKG : G \subset 'N(H) -> coset H @*^-1 (G / H) = H <*> G.
Proof. by move=> nHG; rewrite quotientK ?norm_joinEr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientYK
| |
quotientGKG : H <| G -> coset H @*^-1 (G / H) = G.
Proof. by case/andP; rewrite -{1}ker_coset; apply: morphimGK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientGK
| |
quotient_classx A :
x \in 'N(H) -> A \subset 'N(H) -> x ^: A / H = coset H x ^: (A / H).
Proof. exact: morphim_class. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_class
| |
classes_quotientA :
A \subset 'N(H) -> classes (A / H) = [set xA / H | xA in classes A].
Proof. exact: classes_morphim. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
classes_quotient
| |
cosetpre_set1x :
x \in 'N(H) -> coset H @*^-1 [set coset H x] = H :* x.
Proof. by rewrite -{9}ker_coset; apply: morphpre_set1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_set1
| |
cosetpre_set1_cosetxbar : coset H @*^-1 [set xbar] = xbar.
Proof. by case: (cosetP xbar) => x Nx ->; rewrite cosetpre_set1 ?val_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_set1_coset
| |
cosetpreKC : coset H @*^-1 C / H = C.
Proof. by rewrite /quotient morphpreK ?sub_im_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpreK
| |
trivg_quotient: H / H = 1.
Proof. by rewrite -[X in X / _]ker_coset /quotient morphim_ker. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
trivg_quotient
| |
quotientS1G : G \subset H -> G / H = 1.
Proof. by move=> sGH; apply/trivgP; rewrite -trivg_quotient quotientS. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientS1
| |
sub_cosetpreM : H \subset coset H @*^-1 M.
Proof. by rewrite -{1}ker_coset; apply: ker_sub_pre. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
sub_cosetpre
| |
quotient_properG K :
H <| G -> H <| K -> (G / H \proper K / H) = (G \proper K).
Proof. by move=> nHG nHK; rewrite -cosetpre_proper ?quotientGK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_proper
| |
normal_cosetpreM : H <| coset H @*^-1 M.
Proof. by rewrite -{1}ker_coset; apply: ker_normal_pre. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
normal_cosetpre
| |
cosetpreSKC D :
(coset H @*^-1 C \subset coset H @*^-1 D) = (C \subset D).
Proof. by rewrite morphpreSK ?sub_im_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpreSK
| |
sub_quotient_preA C :
A \subset 'N(H) -> (A / H \subset C) = (A \subset coset H @*^-1 C).
Proof. exact: sub_morphim_pre. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
sub_quotient_pre
| |
sub_cosetpre_quoC G :
H <| G -> (coset H @*^-1 C \subset G) = (C \subset G / H).
Proof. by move=> nHG; rewrite -cosetpreSK quotientGK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
sub_cosetpre_quo
| |
quotient_sub1A : A \subset 'N(H) -> (A / H \subset [1]) = (A \subset H).
Proof.
by move=> nHA /=; rewrite -[gval H in RHS]ker_coset ker_trivg_morphim nHA.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_sub1
| |
quotientSKA B :
A \subset 'N(H) -> (A / H \subset B / H) = (A \subset H * B).
Proof. by move=> nHA; rewrite morphimSK ?ker_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientSK
| |
quotientSGKA G :
A \subset 'N(H) -> H \subset G -> (A / H \subset G / H) = (A \subset G).
Proof. by rewrite -{2}ker_coset; apply: morphimSGK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientSGK
| |
quotient_injG:
{in [pred G : {group gT} | H <| G] &, injective (fun G => G / H)}.
Proof. by rewrite /normal -{1}ker_coset; apply: morphim_injG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_injG
| |
quotient_injG1 G2 :
H <| G1 -> H <| G2 -> G1 / H = G2 / H -> G1 :=: G2.
Proof. by rewrite /normal -[in mem H]ker_coset; apply: morphim_inj. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_inj
| |
quotient_neq1A : H <| A -> (A / H != 1) = (H \proper A).
Proof.
case/andP=> sHA nHA; rewrite /proper sHA -trivg_quotient eqEsubset andbC.
by rewrite quotientS //= quotientSGK.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_neq1
| |
quotient_genA : A \subset 'N(H) -> <<A>> / H = <<A / H>>.
Proof. exact: morphim_gen. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_gen
| |
cosetpre_genC :
1 \in C -> coset H @*^-1 <<C>> = <<coset H @*^-1 C>>.
Proof. by move=> C1; rewrite morphpre_gen ?sub_im_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_gen
| |
quotientRA B :
A \subset 'N(H) -> B \subset 'N(H) -> [~: A, B] / H = [~: A / H, B / H].
Proof. exact: morphimR. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientR
| |
quotient_normA : 'N(A) / H \subset 'N(A / H).
Proof. exact: morphim_norm. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_norm
| |
quotient_normsA B : A \subset 'N(B) -> A / H \subset 'N(B / H).
Proof. exact: morphim_norms. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_norms
| |
quotient_subnormA B : 'N_A(B) / H \subset 'N_(A / H)(B / H).
Proof. exact: morphim_subnorm. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_subnorm
| |
quotient_normalA B : A <| B -> A / H <| B / H.
Proof. exact: morphim_normal. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_normal
| |
quotient_cent1x : 'C[x] / H \subset 'C[coset H x].
Proof.
case Nx: (x \in 'N(H)); first exact: morphim_cent1.
by rewrite coset_default // cent11T subsetT.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_cent1
| |
quotient_cent1sA x : A \subset 'C[x] -> A / H \subset 'C[coset H x].
Proof.
by move=> sAC; apply: subset_trans (quotientS sAC) (quotient_cent1 x).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_cent1s
| |
quotient_subcent1A x : 'C_A[x] / H \subset 'C_(A / H)[coset H x].
Proof. exact: subset_trans (quotientI _ _) (setIS _ (quotient_cent1 x)). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_subcent1
| |
quotient_centA : 'C(A) / H \subset 'C(A / H).
Proof. exact: morphim_cent. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_cent
| |
quotient_centsA B : A \subset 'C(B) -> A / H \subset 'C(B / H).
Proof. exact: morphim_cents. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_cents
| |
quotient_abelianA : abelian A -> abelian (A / H).
Proof. exact: morphim_abelian. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_abelian
| |
quotient_subcentA B : 'C_A(B) / H \subset 'C_(A / H)(B / H).
Proof. exact: morphim_subcent. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_subcent
| |
norm_quotient_preA C :
A \subset 'N(H) -> A / H \subset 'N(C) -> A \subset 'N(coset H @*^-1 C).
Proof.
by move/sub_quotient_pre=> -> /subset_trans-> //; apply: morphpre_norm.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
norm_quotient_pre
| |
cosetpre_normalC D : (coset H @*^-1 C <| coset H @*^-1 D) = (C <| D).
Proof. by rewrite morphpre_normal ?sub_im_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_normal
| |
quotient_normGG : H <| G -> 'N(G) / H = 'N(G / H).
Proof.
case/andP=> sHG nHG.
by rewrite [_ / _]morphim_normG ?ker_coset // im_coset setTI.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_normG
| |
quotient_subnormGA G : H <| G -> 'N_A(G) / H = 'N_(A / H)(G / H).
Proof. by case/andP=> sHG nHG; rewrite -morphim_subnormG ?ker_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_subnormG
| |
cosetpre_cent1x : 'C_('N(H))[x] \subset coset H @*^-1 'C[coset H x].
Proof.
case Nx: (x \in 'N(H)); first by rewrite morphpre_cent1.
by rewrite coset_default // cent11T morphpreT subsetIl.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_cent1
| |
cosetpre_cent1sC x :
coset H @*^-1 C \subset 'C[x] -> C \subset 'C[coset H x].
Proof.
move=> sC; rewrite -cosetpreSK; apply: subset_trans (cosetpre_cent1 x).
by rewrite subsetI subsetIl.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_cent1s
| |
cosetpre_subcent1C x :
'C_(coset H @*^-1 C)[x] \subset coset H @*^-1 'C_C[coset H x].
Proof.
by rewrite -morphpreIdom -setIA setICA morphpreI setIS // cosetpre_cent1.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_subcent1
| |
cosetpre_centA : 'C_('N(H))(A) \subset coset H @*^-1 'C(A / H).
Proof. exact: morphpre_cent. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_cent
| |
cosetpre_centsA C : coset H @*^-1 C \subset 'C(A) -> C \subset 'C(A / H).
Proof. by apply: morphpre_cents; rewrite ?sub_im_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_cents
| |
cosetpre_subcentC A :
'C_(coset H @*^-1 C)(A) \subset coset H @*^-1 'C_C(A / H).
Proof. exact: morphpre_subcent. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_subcent
| |
restrm_quotientEG A (nHG : G \subset 'N(H)) :
A \subset G -> restrm nHG (coset H) @* A = A / H.
Proof. exact: restrmEsub. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
restrm_quotientE
| |
inv_quotient_spec(P : pred {group gT}) : Prop :=
InvQuotientSpec K of Kbar :=: K / H & H \subset K & P K.
|
Variant
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
inv_quotient_spec
| |
inv_quotientS:
Kbar \subset G / H -> inv_quotient_spec (fun K => K \subset G).
Proof.
move=> sKH; exists (coset H @*^-1 Kbar); first by rewrite cosetpreK.
by rewrite sub_cosetpre.
by rewrite sub_cosetpre_quo.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
inv_quotientS
| |
inv_quotientN: Kbar <| G / H -> inv_quotient_spec (fun K => K <| G).
Proof.
move=> nKbar; case/inv_quotientS: (normal_sub nKbar) => K defKbar sHK sKG.
exists K => //; rewrite defKbar -cosetpre_normal !quotientGK // in nKbar.
exact: normalS nHG.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
inv_quotientN
| |
quotientMidrA : A * H / H = A / H.
Proof.
by rewrite [_ /_]morphimMr ?normG //= -!quotientE trivg_quotient mulg1.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientMidr
| |
quotientMidlA : H * A / H = A / H.
Proof.
by rewrite [_ /_]morphimMl ?normG //= -!quotientE trivg_quotient mul1g.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientMidl
| |
quotientYidrG : G \subset 'N(H) -> G <*> H / H = G / H.
Proof.
move=> nHG; rewrite -genM_join quotient_gen ?mul_subG ?normG //.
by rewrite quotientMidr genGid.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientYidr
| |
quotientYidlG : G \subset 'N(H) -> H <*> G / H = G / H.
Proof. by move=> nHG; rewrite joingC quotientYidr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientYidl
| |
quotient_isom: isom G (G / H) (restrm nHG (coset H)).
Proof. by apply/isomP; rewrite ker_restrm setIC ker_coset tiHG im_restrm. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_isom
| |
quotient_isog: isog G (G / H).
Proof. exact: isom_isog quotient_isom. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_isog
| |
coset1_injm: 'injm (@coset gT 1).
Proof. by rewrite ker_coset /=. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset1_injm
| |
quotient1_isom: isom A (A / 1) (coset 1).
Proof. by apply: sub_isom coset1_injm; rewrite ?norms1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient1_isom
| |
quotient1_isog: isog A (A / 1).
Proof. by apply: isom_isog quotient1_isom; apply: norms1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient1_isog
| |
fH:= (coset (f @* H) \o f).
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
fH
| |
quotm_dom_proof: G \subset 'dom fH.
Proof. by rewrite -sub_morphim_pre. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotm_dom_proof
| |
fH_G:= (restrm quotm_dom_proof fH).
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
fH_G
| |
quotm_ker_proof: 'ker (coset H) \subset 'ker fH_G.
Proof.
by rewrite ker_restrm ker_comp !ker_coset morphpreIdom morphimK ?mulG_subr.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotm_ker_proof
| |
quotm:= factm quotm_ker_proof nHG.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotm
| |
quotm_morphism:= [morphism G / H of quotm].
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotm_morphism
| |
quotmEx : x \in G -> quotm (coset H x) = coset (f @* H) (f x).
Proof. exact: factmE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotmE
| |
morphim_quotmA : quotm @* (A / H) = f @* A / f @* H.
Proof. by rewrite morphim_factm [LHS]morphim_restrm morphim_comp morphimIdom. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
morphim_quotm
| |
morphpre_quotmAbar : quotm @*^-1 (Abar / f @* H) = f @*^-1 Abar / H.
Proof.
rewrite morphpre_factm morphpre_restrm morphpre_comp /=.
rewrite morphpreIdom -[Abar / _]quotientInorm quotientK ?subsetIr //=.
rewrite morphpreMl ?morphimS // morphimK // [_ * H]normC ?subIset ?nHG //.
rewrite -quotientE -mulgA quotientMidl /= setIC -morphpreIim setIA.
by rewrite (setIidPl nfHfG) morphpreIim -morphpreMl ?sub1G ?mul1g.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
morphpre_quotm
| |
ker_quotm: 'ker quotm = 'ker f / H.
Proof. by rewrite -morphpre_quotm /quotient morphim1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
ker_quotm
| |
injm_quotm: 'injm f -> 'injm quotm.
Proof. by move/trivgP=> /= kf1; rewrite ker_quotm kf1 quotientE morphim1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
injm_quotm
| |
im_qisom_proof: 'N(H) \subset 'N(G). Proof. by rewrite eqGH. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
im_qisom_proof
| |
qisom_ker_proof: 'ker (coset G) \subset 'ker (coset H).
Proof. by rewrite eqGH. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
qisom_ker_proof
| |
qisom_restr_proof: setT \subset 'N(H) / G.
Proof. by rewrite eqGH im_quotient. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
qisom_restr_proof
| |
qisom:=
restrm qisom_restr_proof (factm qisom_ker_proof im_qisom_proof).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
qisom
| |
qisom_morphism:= Eval hnf in [morphism of qisom].
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
qisom_morphism
| |
qisomEx : qisom (coset G x) = coset H x.
Proof.
case Nx: (x \in 'N(H)); first exact: factmE.
by rewrite !coset_default ?eqGH ?morph1.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
qisomE
| |
val_qisomGx : val (qisom Gx) = val Gx.
Proof.
by case: (cosetP Gx) => x Nx ->{Gx}; rewrite qisomE /= !val_coset -?eqGH.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
val_qisom
| |
morphim_qisomA : qisom @* (A / G) = A / H.
Proof. by rewrite morphim_restrm setTI morphim_factm. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
morphim_qisom
| |
morphpre_qisomA : qisom @*^-1 (A / H) = A / G.
Proof.
rewrite morphpre_restrm setTI morphpre_factm eqGH.
by rewrite morphpreK // im_coset subsetT.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
morphpre_qisom
| |
injm_qisom: 'injm qisom.
Proof. by rewrite -quotient1 -morphpre_qisom morphpreS ?sub1G. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
injm_qisom
| |
im_qisom: qisom @* setT = setT.
Proof. by rewrite -{2}im_quotient morphim_qisom eqGH im_quotient. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
im_qisom
| |
qisom_isom: isom setT setT qisom.
Proof. by apply/isomP; rewrite injm_qisom im_qisom. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
qisom_isom
| |
qisom_isog: [set: coset_of G] \isog [set: coset_of H].
Proof. exact: isom_isog qisom_isom. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
qisom_isog
| |
qisom_inj: injective qisom.
Proof. by move=> x y; apply: (injmP injm_qisom); rewrite inE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
qisom_inj
| |
morphim_qisom_inj: injective (fun Gx => qisom @* Gx).
Proof.
by move=> Gx Gy; apply: injm_morphim_inj; rewrite (injm_qisom, subsetT).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
morphim_qisom_inj
| |
first_isom(G : {group aT}) (f : {morphism G >-> rT}) :
{g : {morphism G / 'ker f >-> rT} | 'injm g &
forall A : {set aT}, g @* (A / 'ker f) = f @* A}.
Proof.
have nkG := ker_norm f.
have skk: 'ker (coset ('ker f)) \subset 'ker f by rewrite ker_coset.
exists (factm_morphism skk nkG) => /=; last exact: morphim_factm.
by rewrite ker_factm -quotientE trivg_quotient.
Qed.
Variables (G H : {group aT}) (f : {morphism G >-> rT}).
Hypothesis sHG : H \subset G.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
first_isom
| |
first_isog: (G / 'ker f) \isog (f @* G).
Proof.
by case: (first_isom f) => g injg im_g; apply/isogP; exists g; rewrite ?im_g.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
first_isog
| |
first_isom_loc: {g : {morphism H / 'ker_H f >-> rT} |
'injm g & forall A : {set aT}, A \subset H -> g @* (A / 'ker_H f) = f @* A}.
Proof.
case: (first_isom (restrm_morphism sHG f)).
rewrite ker_restrm => g injg im_g; exists g => // A sAH.
by rewrite im_g morphim_restrm (setIidPr sAH).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
first_isom_loc
| |
first_isog_loc: (H / 'ker_H f) \isog (f @* H).
Proof.
by case: first_isom_loc => g injg im_g; apply/isogP; exists g; rewrite ?im_g.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
first_isog_loc
| |
second_isom: {f : {morphism H / (K :&: H) >-> coset_of K} |
'injm f & forall A : {set gT}, A \subset H -> f @* (A / (K :&: H)) = A / K}.
Proof.
have ->: K :&: H = 'ker_H (coset K) by rewrite ker_coset setIC.
exact: first_isom_loc.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
second_isom
| |
second_isog: H / (K :&: H) \isog H / K.
Proof. by rewrite setIC -{1 3}(ker_coset K); apply: first_isog_loc. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
second_isog
| |
weak_second_isog: H / (K :&: H) \isog H * K / K.
Proof. by rewrite quotientMidr; apply: second_isog. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
weak_second_isog
| |
homg_quotientS(A : {set gT}) :
A \subset 'N(H) -> A \subset 'N(K) -> H \subset K -> A / K \homg A / H.
Proof.
rewrite -!(gen_subG A) /=; set L := <<A>> => nHL nKL sKH.
have sub_ker: 'ker (restrm nHL (coset H)) \subset 'ker (restrm nKL (coset K)).
by rewrite !ker_restrm !ker_coset setIS.
have sAL: A \subset L := subset_gen A; rewrite -(setIidPr sAL).
rewrite -[_ / H](morphim_restrm nHL) -[_ / K](morphim_restrm nKL) /=.
by rewrite -(morphim_factm sub_ker (subxx L)) morphim_homg ?morphimS.
Qed.
Hypothesis sHK : H \subset K.
Hypothesis snHG : H <| G.
Hypothesis snKG : K <| G.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
homg_quotientS
| |
third_isom: {f : {morphism (G / H) / (K / H) >-> coset_of K} | 'injm f
& forall A : {set gT}, A \subset G -> f @* (A / H / (K / H)) = A / K}.
Proof.
have [[sKG nKG] [sHG nHG]] := (andP snKG, andP snHG).
have sHker: 'ker (coset H) \subset 'ker (restrm nKG (coset K)).
by rewrite ker_restrm !ker_coset subsetI sHG.
have:= first_isom_loc (factm_morphism sHker nHG) (subxx _) => /=.
rewrite ker_factm_loc ker_restrm ker_coset !(setIidPr sKG) /= -!quotientE.
case=> f injf im_f; exists f => // A sAG; rewrite im_f ?morphimS //.
by rewrite morphim_factm morphim_restrm (setIidPr sAG).
Qed.
|
Theorem
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
third_isom
| |
third_isog: (G / H / (K / H)) \isog (G / K).
Proof.
by case: third_isom => f inj_f im_f; apply/isogP; exists f; rewrite ?im_f.
Qed.
|
Theorem
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
third_isog
| |
char_from_quotient(gT : finGroupType) (G H K : {group gT}) :
H <| K -> H \char G -> K / H \char G / H -> K \char G.
Proof.
case/andP=> sHK nHK chHG.
have nsHG := char_normal chHG; have [sHG nHG] := andP nsHG.
case/charP; rewrite quotientSGK // => sKG /= chKG.
apply/charP; split=> // f injf Gf; apply/morphim_fixP => //.
rewrite -(quotientSGK _ sHK); last by rewrite -morphimIim Gf subIset ?nHG.
have{chHG} Hf: f @* H = H by case/charP: chHG => _; apply.
set q := quotm_morphism f nsHG; have{injf}: 'injm q by apply: injm_quotm.
have: q @* _ = _ := morphim_quotm _ _ _; move: q; rewrite Hf => q im_q injq.
by rewrite -im_q chKG // im_q Gf.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
char_from_quotient
|
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