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

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

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

Older notations for the homomorphism h(x) may be x_h, 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 as it is, is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function h : G → H is a group homomorphism if whenever a ∗ b = 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.

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

${\mathop {{\mathrm {ker}}}}(h):=\{u\in G:h(u)=e_{{H}}\}{\mbox{.}}\!$

and the image of h to be

${\mathop {{\mathrm {im}}}}(h):=h(G):=\left\{h(u)\colon u\in G\right\}{\mbox{.}}\!$

The kernel of h is a normal subgroup of G and the image of h is a subgroup of H:

$h\left(g^{{-1}}\circ u\circ g\right)=h(g)^{{-1}}\cdot h(u)\cdot h(g)=h(g)^{{-1}}\cdot e_{H}\cdot h(g)=h(g)^{{-1}}\cdot h(g)=e_{H}.$

The homomorphism h is injective (and called a group monomorphism) if and only if ker(h) = {e_G}.

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.

Examples

Consider the cyclic group Z/3Z = {0, 1, 2} and the group of integers Z with addition. 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.

Consider the group

$G:=\left\{{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\bigg |}a>0,b\in {\mathbf {R}}\right\}$

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 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 category of groups

If h : G → H and k : H → K are group homomorphisms, then so is k o h : G → K. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.

Types of homomorphic maps

If the homomorphism h is a bijection, then one can show that its inverse is also a group homomorphism, and h is called a group isomorphism; 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.

If h: G → G is a group homomorphism, we call it an endomorphism of G. If furthermore it is bijective and hence an isomorphism, it is called an automorphism. 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.

An epimorphism is a surjective homomorphism, that is, a homomorphism which is onto as a function. A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a function.

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) o f = (h o f) + (k o f) and g o (h + k) = (g o h) + (g o k).

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

Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

External links

Group Homomorphism, PlanetMath.org.

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