Group homomorphism

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

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

Now by multiplying for the inverse of ${\displaystyle h(u)}$ (or applying the cancellation rule) we obtain

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

Similarly,

${\displaystyle e_{H}=h(e_{G})=h(u*u^{-1})=h(u)\cdot h(u^{-1})}$

Therefore for the uniqueness of the inverse: ${\displaystyle h(u^{-1})=h(u)^{-1}}$.

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 which are mapped to the identity in H

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

and the image of h to be

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

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

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

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

${\displaystyle {\begin{aligned}&&h(g_{1})&=h(g_{2})\\\Leftrightarrow &&h(g_{1})\cdot h(g_{2})^{-1}&=e_{H}\\\Leftrightarrow &&h\left(g_{1}\circ g_{2}^{-1}\right)&=e_{H},\ \operatorname {ker} (h)=\{e_{G}\}\\\Rightarrow &&g_{1}\circ g_{2}^{-1}&=e_{G}\\\Leftrightarrow &&g_{1}&=g_{2}\end{aligned}}}$

• 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 which are divisible by 3.

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

• Homomorphism
• Fundamental theorem on homomorphisms
• Quasimorphism
• Ring homomorphism