Group homomorphism

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

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

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_H of H,

\(h(e_G) = e_H\)

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".

In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map that 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.

Let \(e_{H}\) be the identity element of the group (H, ·) and \(u \in G\), then

\(h(u) \cdot e_{H} = h(u) = h(u*e_{G}) = h(u) \cdot h(e_{G})\)

Now by multiplying by the inverse of \(h(u)\) (or applying the cancellation rule) we obtain

\(e_{H} = h(e_{G})\)

Similarly,

\(e_H = h(e_G) = h(u*u^{-1}) = h(u)\cdot h(u^{-1})\)

Therefore, by the uniqueness of the inverse: \(h(u^{-1}) = h(u)^{-1}\).

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, +).

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

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

and the image of h to be

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

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

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

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:

\(\begin{aligned} && h(g_1) &= h(g_2) \\ \Leftrightarrow && h(g_1) \cdot h(g_2)^{-1} &= e_H \\ \Leftrightarrow && h\left(g_1 \circ g_2^{-1}\right) &= e_H,\ \operatorname{ker}(h) = \{e_G\} \\ \Rightarrow && g_1 \circ g_2^{-1} &= e_G \\ \Leftrightarrow && g_1 &= g_2 \end{aligned}\)

• 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

  \(G \equiv \left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \bigg| a > 0, b \in \mathbf{R}\right\}\)

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

  \(\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \mapsto a^u\)

  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

  \(f_u(a) = a^u\)

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