# Group homomorphism

__: Mathematical function between groups that preserves multiplication structure__

**Short description**Algebraic structure → Group theoryGroup theory |
---|

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

- [math]\displaystyle{ h(u*v) = h(u) \cdot h(v) }[/math]

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*,

- [math]\displaystyle{ h(e_G) = e_H }[/math]

and it also maps inverses to inverses in the sense that

- [math]\displaystyle{ h\left(u^{-1}\right) = h(u)^{-1}. \, }[/math]

Hence one can say that *h* "is compatible with the group structure".

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

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

*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.

## Types

- Monomorphism
- A group homomorphism that is injective (or, one-to-one); i.e., preserves distinctness.
- Epimorphism
- A group homomorphism that is surjective (or, onto); i.e., reaches every point in the codomain.
- Isomorphism
- A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. 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. - Endomorphism
- A group homomorphism,
*h*:*G*→*G*; the domain and codomain are the same. Also called an endomorphism of*G*. - Automorphism
- A group endomorphism that is bijective, and hence an isomorphism. The set of all automorphisms of a group
*G*, with functional composition as operation, itself forms 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**, +).

## Image and kernel

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

- [math]\displaystyle{ \operatorname{ker}(h) := \left\{u \in G\colon h(u) = e_{H}\right\}. }[/math]

and the *image of h* to be

- [math]\displaystyle{ \operatorname{im}(h) := h(G) \equiv \left\{h(u)\colon u \in G\right\}. }[/math]

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*:

- [math]\displaystyle{ \begin{align} h\left(g^{-1} \circ u \circ g\right) &= h(g)^{-1} \cdot h(u) \cdot h(g) \\ &= h(g)^{-1} \cdot e_H \cdot h(g) \\ &= h(g)^{-1} \cdot h(g) = e_H, \end{align} }[/math]

and the image of h is a subgroup of *H*.

The homomorphism, *h*, is a *group monomorphism*; i.e., *h* is injective (one-to-one) if and only if ker(*h*) = {*e*_{G}}. Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:

- [math]\displaystyle{ \begin{align} && h(g_1) &= h(g_2) \\ \Leftrightarrow && h(g_1) \cdot h(g_2)^{-1} &= e_H \\ \Leftrightarrow && h\left(g_1 \circ g_2^{-1}\right) &= e_H,\ \operatorname{ker}(h) = \{e_G\} \\ \Rightarrow && g_1 \circ g_2^{-1} &= e_G \\ \Leftrightarrow && g_1 &= g_2 \end{align} }[/math]

## Examples

- Consider the cyclic group Z
_{3}= (**Z**/3**Z**, +) = ({0, 1, 2}, +) and the group of integers (**Z**, +). 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.

- The set
- [math]\displaystyle{ G \equiv \left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \bigg| a \gt 0, b \in \mathbf{R}\right\} }[/math]

forms a group under matrix multiplication. For any complex number

*u*the function*f*:_{u}*G*→**C**defined by^{*}- [math]\displaystyle{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \mapsto a^u }[/math]

- Consider multiplicative group of positive real numbers (
**R**^{+}, ⋅) for any complex number*u*the function*f*:_{u}**R**^{+}→**C**defined by- [math]\displaystyle{ f_u(a) = a^u }[/math]

- 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 function [math]\displaystyle{ \Phi: (\mathbb{N}, +) \rightarrow (\mathbb{R}, +) }[/math], defined by [math]\displaystyle{ \Phi(x) = \sqrt[]{2}x }[/math] is a homomorphism.

## Category of groups

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.

## 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*) ∘*f*= (*h*∘*f*) + (*k*∘*f*) and*g*∘ (*h*+*k*) = (*g*∘*h*) + (*g*∘*k*).

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.

## See also

## References

- Dummit, D. S.; Foote, R. (2004).
*Abstract Algebra*(3rd ed.). Wiley. pp. 71–72. ISBN 978-0-471-43334-7. - Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics,**211**(Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4

## External links

- Rowland, Todd. "Group Homomorphism". http://mathworld.wolfram.com/GroupHomomorphism.html.

Original source: https://en.wikipedia.org/wiki/Group homomorphism.
Read more |