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image of an abelian group is abelian

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This commit is contained in:
Egbert Rijke 2016-11-03 23:30:44 -04:00
parent 699531a74c
commit 7ab6eafc3c
1 changed files with 28 additions and 6 deletions
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34
algebra/subgroup.hlean
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@ -123,12 +123,11 @@ namespace group
(is_normal_subgroup : is_normal R)
attribute subgroup_rel.R [coercion]
abbreviation subgroup_to_rel [parsing_only] [unfold 2] := @subgroup_rel.R
abbreviation subgroup_has_one [parsing_only] [unfold 2] := @subgroup_rel.Rone
abbreviation subgroup_respect_mul [parsing_only] [unfold 2] := @subgroup_rel.Rmul
abbreviation subgroup_respect_inv [parsing_only] [unfold 2] := @subgroup_rel.Rinv
abbreviation is_normal_subgroup [parsing_only] [unfold 2] :=
@normal_subgroup_rel.is_normal_subgroup
abbreviation subgroup_to_rel [unfold 2] := @subgroup_rel.R
abbreviation subgroup_has_one [unfold 2] := @subgroup_rel.Rone
abbreviation subgroup_respect_mul [unfold 2] := @subgroup_rel.Rmul
abbreviation subgroup_respect_inv [unfold 2] := @subgroup_rel.Rinv
abbreviation is_normal_subgroup [unfold 2] := @normal_subgroup_rel.is_normal_subgroup
variables {G G' : Group} (H : subgroup_rel G) (N : normal_subgroup_rel G) {g g' h h' k : G}
{A B : CommGroup}
@ -258,6 +257,29 @@ namespace group
definition image {G H : Group} (f : G →g H) : Group :=
subgroup (image_subgroup f)
definition CommGroup_of_Group.{u} (G : Group.{u}) (H : Π (g h : G), mul g h = mul h g) : CommGroup.{u} :=
begin
induction G,
induction struct,
fapply CommGroup.mk,
exact carrier,
fapply comm_group.mk,
repeat assumption,
exact H
end
definition comm_image {G : CommGroup} {H : Group} (f : G →g H) : CommGroup :=
CommGroup_of_Group (image f)
begin
intro g h,
induction g with x t, induction h with y s,
fapply subtype_eq,
induction t with p, induction s with q, induction p with g p, induction q with h q, induction p, induction q,
refine (((respect_mul f g h)⁻¹ ⬝ _) ⬝ (respect_mul f h g)),
apply (ap f),
induction G, induction struct, apply mul_comm
end
definition image_incl {G H : Group} (f : G →g H) : image f →g H :=
incl_of_subgroup (image_subgroup f)