Group homomorphism

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 (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/2Z, +).

Image and kernel

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

${\displaystyle \ker (h):=\left\{u\in G:h(u)=e_H\right\}.}$

and the image of h to be

${\displaystyle \mathrm{im}(h):=h(G)\equiv \left\{h(u):u\in G\right\}.}$

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. Assume ${\displaystyle u\in \ker (h)}$ and show ${\displaystyle g^{-1}\circ u\circ g\in \ker (h)}$ for arbitrary ${\displaystyle u,g}$:

${\displaystyle \begin{array}{cc}h\left(g^{-1}\circ u\circ g\right)& =h(g{)}^{-1}\cdot h(u)\cdot h(g)\\ & =h(g{)}^{-1}\cdot e_H\cdot h(g)\\ & =h(g{)}^{-1}\cdot h(g)=e_H,\end{array}}$

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}. Injectivity directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injectivity:

${\displaystyle \begin{array}{cccc}& & h(g_1)& =h(g_2)\\ \iff & & h(g_1)\cdot h(g_2{)}^{-1}& =e_H\\ \iff & & h\left(g_1\circ g_2^{-1}\right)& =e_H,\ker (h)=\{e_G\}\\ \Rightarrow & & g_1\circ g_2^{-1}& =e_G\\ \iff & & g_1& =g_2\end{array}}$

Examples

• Consider the cyclic group Z₃ = (Z/3Z, +) = ({0, 1, 2}, +) and the group of integers (Z, +). The map h : Z → Z/3Z with h(u) = u mod 3 is a group homomorphism. It is surjective and its kernel consists of all integers that are divisible by 3.

• The set

  ${\displaystyle G\equiv \left\{\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right)|a>0,b\in \mathbf{R}\right\}}$

  forms a group under matrix multiplication. For any complex number u, the function f_u : G → C^* defined by

  ${\displaystyle \left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right)\mapsto a^u}$

  is a group homomorphism.
• Consider a multiplicative group of positive real numbers (R⁺, ⋅). For any complex number u, the function f_u : R⁺ → C^* defined by

  ${\displaystyle f_u(a)=a^u}$

  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 : 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 ${\displaystyle \mathrm{\Phi }:(\mathbb{Z},+)\to (\mathbb{R},+)}$, defined by ${\displaystyle \mathrm{\Phi }(x)=\sqrt[]{2}x}$ is a homomorphism.
• Consider the two groups ${\displaystyle ({\mathbb{R}}^{+},\ast )}$ and ${\displaystyle (\mathbb{R},+)}$, represented respectively by ${\displaystyle G}$ and ${\displaystyle H}$, where ${\displaystyle {\mathbb{R}}^{+}}$ is the positive real numbers. Then, the function ${\displaystyle f:G\to H}$ defined by the logarithm function 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 (specifically the category of groups).

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/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.