# Group homomorphism

In mathematics, given two groups, (*G*, ∗) and (*H*, ·), a **group homomorphism** from (*G*, ∗) to (*H*, ·) is a function *h* : *G* → *H* such that for all *u* and *v* in *G* it holds that

where the group operation on the left side of the equation is that of *G* and on the right side that of *H*.

From this property, one can deduce that *h* maps the identity element *e _{G}* of

*G*to the identity element

*e*of

_{H}*H*,

Older notations for the homomorphism *h*(*x*) may be *x*^{h} or *x*_{h},^{[citation needed]} though this may be confused as an index or a general subscript. In automata theory, sometimes homomorphisms are written to the right of their arguments without parentheses, so that *h*(*x*) becomes simply *x h*.^{[citation needed]}

In areas of mathematics where one considers groups endowed with additional structure, a *homomorphism* sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.

The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function *h* : *G* → *H* is a group homomorphism if whenever

In other words, the group *H* in some sense has a similar algebraic structure as *G* and the homomorphism *h* preserves that.

We define the *kernel of h* to be the set of elements in *G* which are mapped to the identity in *H*

The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The first isomorphism theorem states that the image of a group homomorphism, *h*(*G*) is isomorphic to the quotient group *G*/ker *h*.

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

If *h* : *G* → *H* and *k* : *H* → *K* are group homomorphisms, then so is *k* ∘ *h* : *G* → *K*. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.

If *G* and *H* are abelian (i.e., commutative) groups, then the set Hom(*G*, *H*) of all group homomorphisms from *G* to *H* is itself an abelian group: the sum *h* + *k* of two homomorphisms is defined by

The commutativity of *H* is needed to prove that *h* + *k* is again a group homomorphism.

The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if *f* is in Hom(*K*, *G*), *h*, *k* are elements of Hom(*G*, *H*), and *g* is in Hom(*H*, *L*), then

Since the composition is associative, this shows that the set End(*G*) of all endomorphisms of an abelian group forms a ring, the *endomorphism ring* of *G*. For example, the endomorphism ring of the abelian group consisting of the direct sum of *m* copies of **Z**/*n***Z** is isomorphic to the ring of *m*-by-*m* matrices with entries in **Z**/*n***Z**. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category.