Algebraic structure → Group theoryGroup theory |
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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

and it also maps inverses to inverses in the sense that

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.

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.

- 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 (except of identity element) and are identical for all practical purposes. I.e. we re-label all elements except identity. - 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**, +).

Main articles: Image (mathematics) and kernel (algebra) |

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 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*.

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:

- 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
forms a group under matrix multiplication. For any complex number

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

- 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 , defined by is a homomorphism.
- Consider the two groups and , represented respectively by and , where is the positive real numbers. Then, the function defined by the logarithm function is a homomorphism.

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

- (
*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.