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:$h(u*v)\; =\; h(u)\; -\; h(v)$

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 theirarguments, 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.

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":ker("h") = { "u" in "G" : "h"("u") = "e_H" } and the "image of h" to be:im("h") = { "h"("u") : "u" in "G" }.The kernel is a normal subgroup of "G" (in fact, "h"("g"^-1 "u" "g") = "h"("g")^-1 "h"("u") "h"("g") = "h"("g")^-1 "e_H" "h"("g") = "h"("g")^-1 "h"("g") = "e_H") and the image is a subgroup of "H".The homomorphism "h" is injective (and called a "group monomorphism") if and only if ker("h") = {"e"_"G"}.

The kernel and image nowrap begin"a"("G") = {"a"("g"), "g" ∈ "G"}nowrap end 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, "a"("G") is isomorphic to the quotient group "G"/ker "a".

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.

* 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 { 2π"ki" : "k" in Z }, as can be seen from Euler's formula.

* Given any two groups "G" and "H", the map "h" : "G" → "H" which sends every element of "G" to the identity element of "H" is a homomorphism; its kernel is all of "G".

* Given any group "G", the identity map id : "G" → "G" with id("u") = "u" for all "u" in "G" is a group homomorphism.

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.

ee also

*Fundamental theorem on homomorphisms

References

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External links

*planetmath reference|id=719|title=Group Homomorphism