# Group homomorphism

### From Online Dictionary of Crystallography

Homomorphisme de groupes (*Fr*). Gruppenhomomorphismus (*Ge*). Homomorfismo de grupos (*Sp*). Omomorfismo di gruppi (*It*). 準同形 (*Ja*).

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## Homomorphism between groups

A **group homomorphism** from a group (*G*, *) to a group (*H*, #) is a mapping *f* : *G* → *H* that preserves the composition law, *i.e.* for all *u* and *v* in *G* one has:

*f*(*u* * *v*) = *f*(*u*) # *f*(*v*).

A homomorphism *f* maps the identity element 1_{G} of *G* to the identity element 1_{H} of *H*, and it also maps inverses to inverses: *f*(*u*^{−1}) = *f*(*u*)^{−1}.

## Kernel and image

The *kernel* of the homomorphism *f* is the set of elements of *G* that are mapped to the identity of *H*:

ker(*f*) = { *u* in *G* : *f*(*u*) = 1_{H} }.

The *image* of the homomorphism *f* is the subset of elements of *H* to which at least one element of *G* is mapped by *f*:

im(*f*) = { *f*(*u*) : *u* in *G* }.

The kernel is a normal subgroup of *G* and the image is a subgroup of *H*.

## Types of homomorphisms

Homomorphisms can be classified according to different criteria, among which are the relation between *G* and *H* and the nature of the mapping.

### Surjective, injective and bijective homomorphisms

An **epimorphism** is a surjective homomorphism, that is, a homomorphism which is *onto* as a mapping. The image of the homomorphism is the whole of *H*, *i.e.* im(*f*) = *H*.

A **monomorphism** is an injective homomorphism, that is, a homomorphism which is *one-to-one* as a mapping. In this case, ker( *f* ) = {1_{G} }.

If the homomorphism *f* is a bijection, then its inverse is also a group homomorphism, and *f* is called an ** isomorphism**; the groups (*G*,*) and (*H*,#) are called *isomorphic* and differ only in the notation of their elements (and possibly their binary operations), while they can be regarded as identical for most practical purposes.

### Homomorphisms from a group to itself (G = H)

An **endomorphism** is a homomorphism of a group to itself: *f* : *G* → *G*.

A bijective (invertible) endomorphism (which is hence an isomorphism) is called an **automorphism**. The kernel of the automorphism is the identity of *G* (1_{G}) and the image of the automorphism coincides with *G*. The set of all automorphisms of a group (*G*,*) forms itself a group, the *automorphism group* of *G*, **Aut( G)**.