# Tour:Homomorphism of groups

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WHAT YOU NEED TO DO:

- Read, and understand, the definition of homomorphism of groups.
- Try proving that a composite of homomorphisms is a homomorphism and the inverse of a bijective homomorphism is a homomorphism.
- Read, and understand, the definitions of four special kinds of homomorphisms. You've already seen one of these.

## Contents

## Definition

### Textbook definition (with symbols)

Let and be groups. Then a map is termed a **homomorphism** of groups if satisfies the following condition:

for all in

### Universal algebraic definition (with symbols)

Let and be groups. Then a map is termed a **homomorphism** of groups if satisfies *all* the following conditions:

- for all in

### Equivalence of definitions

`For full proof, refer: Equivalence of definitions of homomorphism of groups`

The textbook definition and universal algebraic definition of homomorphism of groups are equivalent. In other words, for a map between groups to be a homomorphism of groups, it suffices to check that it preserves the binary operation.

## Facts

### Composition

If and are homomorphisms, then the composite mapping is a homomorphism from to . This follows directly from either definition.

### Inverse

If a homomorphism is bijective, then its set-theoretic inverse map is also a homomorphism.

## Related terms

### Endomorphism

`Further information: endomorphism`

An endomorphism of a group is a homomorphism from the group to itself. Note that every group always has the following two endomorphisms:

- The trivial endomorphism that sends every element of the group to the identity element
- The identity map that sends every element to itself

The endomorphisms of a group form a monoid, termed the endomorphism monoid, under composition.

### Isomorphism

`Further information: isomorphism of groups`

An isomorphism of groups is a bijective homomorphism from one to the other. Equivalently it is a homomorphism for which an inverse map exists, which is also a homomorphism.

If there exists an isomorphism between two groups, they are termed isomorphic groups. Isomorphic groups are equivalent with respect to all group-theoretic constructions. In fact, the isomorphism gives the equivalence.

### Automorphism

`Further information: automorphism of a group`

An automorphism of a group is an isomorphism of the group with itself. Equivalently, it is a bijective endomorphism from the group to itself.

Note that every injective endomorphism may not be an automorphism. Similarly, any surjective endomorphism may not be an automorphism.

Automorphisms of groups can be viewed as symmetries of the group structure. The collection of automorphisms of a group forms a group under composition and this is termed the automorphism group of the given group.

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Group action|UP: Introduction five (beginners)|NEXT: Equivalence of definitions of homomorphismExpected time for this page: 20 minutes

General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part