# Trivial group homomorphisms

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-08-21.

module group-theory.trivial-group-homomorphisms where

Imports
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.propositions
open import foundation.sets
open import foundation.universe-levels

open import group-theory.groups
open import group-theory.homomorphisms-groups


## Idea

A trivial group homomorphism from G to H is a group homomorphism f : G → H such that f x ＝ 1 for every x : G.

## Definitions

### The predicate of being a trivial group homomorphism

module _
{l1 l2 : Level} (G : Group l1) (H : Group l2) (f : hom-Group G H)
where

is-trivial-prop-hom-Group : Prop (l1 ⊔ l2)
is-trivial-prop-hom-Group =
Π-Prop
( type-Group G)
( λ x → Id-Prop (set-Group H) (map-hom-Group G H f x) (unit-Group H))

is-trivial-hom-Group : UU (l1 ⊔ l2)
is-trivial-hom-Group = type-Prop is-trivial-prop-hom-Group

is-prop-is-trivial-hom-Group : is-prop is-trivial-hom-Group
is-prop-is-trivial-hom-Group = is-prop-type-Prop is-trivial-prop-hom-Group


### The trivial group homomorphism

module _
{l1 l2 : Level} (G : Group l1) (H : Group l2)
where

trivial-hom-Group : hom-Group G H
pr1 trivial-hom-Group x = unit-Group H
pr2 trivial-hom-Group = inv (left-unit-law-mul-Group H (unit-Group H))