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 *h*(*u* * *v*) = *h*(*u*) · *h*(*v*) From this property, one can deduce that *h* maps the identity element *e*_{G} of *G* to the identity element *e*_{H} of *H*, and it also maps inverses to inverses in the sense that *h*(*u*^{-1}) = *h*(*u*)^{-1}. Hence one can say that *h* "is compatible with the group structure". Older notations for the homomorphism *h*(*x*) may be *x*_{h}, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that *h*(*x*) becomes simply *x h*. This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right. 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. ## Image and kernel
We define the *kernel of h* to be - ker(
*h*) = { *u* in *G* : *h*(*u*) = *e*_{H} } and the *image of h* to be - im(
*h*) = { *h*(*u*) : *u* in *G* }. The kernel is a normal subgroup of *G* (in fact, *h*(*g*^{-1} *u* *g*) = *h*(*g*)^{-1} *e*_{H} *h*(*g*) = *h*(*g*)^{-1} *h*(*g*) = *e*_{H}) and the image is a subgroup of *H*. The homomorphism *h* is injective (and called a *group monomorphism*) if and only if ker(*h*) = {*e*_{G}}.
## Examples - Consider the cyclic group
**Z**/3**Z** = {0, 1, 2} and the group of integers **Z** with addition. The map *h* : **Z** `->` **Z**/3**Z** with *h*(*u*) = *u* modulo 3 is a group homorphism (see modular arithmetic). It is surjective and its kernel consists of all integers which are divisible by 3. - The exponential map yields a group homorphism from the group of real numbers
**R** with addition to the group of non-zero real numbers **R**^{*} with multiplication. The kernel is {0} and the image consists of the positive real numbers. - The exponential map also yields a group homomorphism from the group of complex numbers
**C** with addition to the group of non-zero complex numbers **C**^{*} with multiplication. This map is surjective and has the kernel { 2π*ki* : *k* in **Z** }, as can be seen from Euler's formula. - Given any two groups
*G* and *H*, the map *h* : *G* `->` *H* which sends every element of *G* to the identity element of *H* is a homomorphism; its kernel is all of *G*. - Given any group
*G*, the identity map id : *G* `->` *G* with id(*u*) = *u* for all *u* in *G* is a group homomorphism. ## The category of groups If *h* : *G* `->` *H* and *k* : *H* `->` *K* are group homomorphisms, then so is *k* o *h* : *G* `->` *K*. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.
## Isomorphisms, endomorphisms and automorphisms If the homomorphism *h* is a bijection, then one can show that its inverse is also a group homomorphism, and *h* is called a *group isomorphism*; in this case, the groups *G* and *H* are called *isomorphic*: they differ only in the notation of their elements and are identical for all practical purposes. If *h*: *G* `->` *G* is a group homomorphism, we call it an *endomorphism* of *G*. If furthermore it is bijective and hence an isomorphism, it is called an *automorphism*. The set of all automorphisms of a group *G*, with functional composition as operation, forms itself a group, the *automorphism group* of *G*. It is denoted by Aut(*G*). As an example, the automorphism group of (**Z**, +) contains only two elements, the identity and multiplication with -1; it is isomorphic to **Z**/2**Z**.
## Homomorphisms of abelian groups 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 - (
*h* + *k*)(*u*) = *h*(*u*) + *k*(*u*) for all *u* in *G*. 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 - (
*h* + *k*) o *f* = (*h* o *f*) + (*k* o *f*) and *g* o (*h* + *k*) = (*g* o *h*) + (*g* o *k*). 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 two copies of **Z**/2**Z** (the Klein four-group) is isomorphic to the ring of 2-by-2 matrices with entries in **Z**/2**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. |