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 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.

Properties

Let $e_{H}$ be the identity element of the (H, ·) group 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 for 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 for the uniqueness of the inverse: $h(u^{-1})=h(u)^{-1}$ .

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

{\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}}

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 which 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. Then 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 $\Phi :(\mathbb {N} ,+)\rightarrow (\mathbb {R} ,+)$ , 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.

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.

References

Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3rd ed.). Wiley. pp. 71–72. ISBN 978-0-471-43334-7.
Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001

External links

Rowland, Todd & Weisstein, Eric W. "Group Homomorphism". MathWorld.