# Library Algebra.Group_of_group_hom

Set Implicit Arguments.
Unset Strict Implicit.
Require Export Group_util.
Require Export Abelian_group_facts.
Title "The abelian group of morphisms from a group to an abelian group."
Section Def.
Variable G : GROUP.
Variable G' : ABELIAN_GROUP.

Definition group_hom_law : f g : Hom G G', Hom G G'.
intros f0 g.
apply
(BUILD_HOM_GROUP (G:=G) (G':=G')
(ff:=fun x : Gsgroup_law G' (f0 x) (g x))).
auto with algebra.
intros x y; try assumption.
apply
Trans
with
(sgroup_law G'
(sgroup_law G' (Ap (sgroup_map (monoid_sgroup_hom f0)) x)
(Ap (sgroup_map (monoid_sgroup_hom f0)) y))
(sgroup_law G' (Ap (sgroup_map (monoid_sgroup_hom g)) x)
(Ap (sgroup_map (monoid_sgroup_hom g)) y)));
auto with algebra.

apply Trans with (sgroup_law G' (monoid_unit G') (monoid_unit G'));
auto with algebra.
Defined.

Definition group_hom_unit : Hom G G'.
apply (BUILD_HOM_GROUP (G:=G) (G':=G') (ff:=fun x : Gmonoid_unit G')).
auto with algebra.
auto with algebra.
auto with algebra.
Defined.

Definition group_hom_inv : f : Hom G G', Hom G G'.
intros f0.
apply
(BUILD_HOM_GROUP (G:=G) (G':=G') (ff:=fun x : Ggroup_inverse G' (f0 x))).
auto with algebra.
intros x y; try assumption.
apply
Trans
with
(Ap (sgroup_map (monoid_sgroup_hom f0))
(group_inverse G (sgroup_law G x y))); auto with algebra.
apply
Trans
with
(Ap (sgroup_map (monoid_sgroup_hom f0))
(sgroup_law G (group_inverse G y) (group_inverse G x)));
auto with algebra.
apply
Trans
with
(sgroup_law G'
(Ap (sgroup_map (monoid_sgroup_hom f0)) (group_inverse G y))
(Ap (sgroup_map (monoid_sgroup_hom f0)) (group_inverse G x)));
auto with algebra.
apply
Trans
with
(sgroup_law G'
(Ap (sgroup_map (monoid_sgroup_hom f0)) (group_inverse G x))
(Ap (sgroup_map (monoid_sgroup_hom f0)) (group_inverse G y)));
auto with algebra.
apply
Trans
with
(Ap (sgroup_map (monoid_sgroup_hom f0)) (group_inverse G (monoid_unit G)));
auto with algebra.
apply Trans with (Ap (sgroup_map (monoid_sgroup_hom f0)) (monoid_unit G));
auto with algebra.
Defined.

Definition group_hom : ABELIAN_GROUP.
apply
(BUILD_ABELIAN_GROUP (E:=Hom G G') (genlaw:=group_hom_law)
(e:=group_hom_unit) (geninv:=group_hom_inv)).
intros x x' y y' H' H'0; try assumption.
simpl in |- ×.
red in |- ×.
simpl in |- ×.
auto with algebra.
intros x y z; try assumption.
simpl in |- ×.
red in |- ×.
simpl in |- ×.
auto with algebra.
intros x; try assumption.
simpl in |- ×.
red in |- ×.
simpl in |- ×.
auto with algebra.
intros x y H'; try assumption.
simpl in |- ×.
red in |- ×.
simpl in |- ×.
auto with algebra.
intros x; try assumption.
simpl in |- ×.
red in |- ×.
simpl in |- ×.
auto with algebra.
intros x y; try assumption.
simpl in |- ×.
red in |- ×.
simpl in |- ×.
auto with algebra.
Defined.

Lemma group_hom_law_prop :
(f g : group_hom) (x : G),
Equal (sgroup_law _ f g x) (sgroup_law _ (f x) (g x)).
simpl in |- *; auto with algebra.
Qed.

Lemma group_hom_unit_prop :
x : G, Equal (monoid_unit group_hom x) (monoid_unit G').
simpl in |- *; auto with algebra.
Qed.

Lemma group_hom_inv_prop :
(f : group_hom) (x : G),
Equal (group_inverse group_hom f x) (group_inverse G' (f x)).
simpl in |- *; auto with algebra.
Qed.
End Def.
Hint Resolve group_hom_law_prop group_hom_unit_prop group_hom_inv_prop:
algebra.