# Undergraduate Mathematics/Group homomorphism

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 hand side of the equation is that of *G* and on the right hand 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*, and it also maps inverses to inverses in the sense that

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.

## Intuition[edit]

The purpose of defining a group homomorphism as it is, 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 *a* ∗ *b* = *c* we have *h*(*a*) ⋅ *h*(*b*) = *h*(*c*). In other words, the group *H* in some sense has a similar algebraic structure as *G* and the homomorphism *h* preserves that.

## Image and kernel[edit]

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

and the *image of h* to be

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

The homomorphism *h* is injective (and called a *group monomorphism*) if and only if ker(*h*) = {*e*_{G}}.

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*.

## Examples[edit]

- 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*mod 3 is a group homomorphism. It is surjective and its kernel consists of all integers which are divisible by 3.

- Consider the group

- For any complex number
*u*the function*f*:_{u}*G*→**C**defined by:

- is a group homomorphism.

- Consider multiplicative group of positive real numbers (
**R**^{+}, ⋅) for any complex number*u*the function*f*:_{u}**R**^{+}→**C**defined by:

- is a group homomorphism.

- The exponential map yields a group homomorphism 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*∈**Z**}, as can be seen from Euler's formula. Fields like**R**and**C**that have homomorphisms from their additive group to their multiplicative group are thus called exponential fields.

## The category of groups[edit]

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.

## Types of homomorphic maps[edit]

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 transformation and multiplication with −1; it is isomorphic to **Z**/2**Z**.

An **epimorphism** is a surjective homomorphism, that is, a homomorphism which is *onto* as a function. A **monomorphism** is an injective homomorphism, that is, a homomorphism which is *one-to-one* as a function.

## Homomorphisms of abelian groups[edit]

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 *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.