# Group homomorphism

Image of a group homomorphism (h) from G (left) to H (right). The smaller oval inside H is the image of h. N is the kernel of h and aN is a coset of N.

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

$h(u*v) = h(u) \cdot h(v)$

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 eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that

$h\left(u^{-1}\right) = h(u)^{-1}. \,$

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

Older notations for the homomorphism h(x) may be xh, 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

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 : GH is a group homomorphism if whenever ab = 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 of group homomorphism

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 homomorphism, h: GG; the domain and codomain are the same. Also called an endomorphism of G.
Automorphism
An endomorphism that is bijective, and hence an isomorphism. 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/2Z.

## 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

$\operatorname{ker}(h) \equiv \left\{u \in G\colon h(u) = e_{H}\right\}.$

and the image of h to be

$\operatorname{im}(h) \equiv h(G) \equiv \left\{h(u)\colon u \in G\right\}.$

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:

\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}

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

\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}

## Examples

• Consider the cyclic group Z/3Z = {0, 1, 2} and the group of integers Z with addition. The map h : ZZ/3Z 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
$G \equiv \left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \bigg| a > 0, b \in \mathbf{R}\right\}$

For any complex number u the function fu : GC defined by:

$\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \mapsto a^u$
is a group homomorphism.
• Consider multiplicative group of positive real numbers (R+, ⋅) for any complex number u the function fu : R+C defined by:
$f_u(a) = a^u$
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 {2πki : kZ}, 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

If h : GH and k : HK are group homomorphisms, then so is kh : GK. 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 = (hf) + (kf)    and    g ∘ (h + k) = (gh) + (gk).

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/nZ is isomorphic to the ring of m-by-m matrices with entries in Z/nZ. 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.