Group homomorphism

Depiction of a group homomorphism (h) from G (left) to H (right). The oval inside H is the image of h. N is the kernel of h and aN is a coset of N.

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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

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,

and it also maps inverses to inverses in the sense that

<math> h\left(u^{-1}\right) = h(u)^{-1}. \,</math>

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.

PropertiesEdit

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

Now by multiplying for the inverse of <math>h(u)</math> (or applying the cancellation rule) we obtain

Similarly,

Therefore for the uniqueness of the inverse: <math>h(u^{-1}) = h(u)^{-1}</math>.

TypesEdit

Monomorphism Template:Anchor: 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 kernelEdit

Template:Main article We define the kernel of h to be the set of elements in G which are mapped to the identity in H

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

and the image of h to be

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

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 <math>u \in \operatorname{ker}(h)</math> and show <math>g^{-1} \circ u \circ g \in \operatorname{ker}(h)</math> for arbitrary <math>u, g</math>:

<math>\begin{align}

 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{align}</math> 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 Template:Nowrap}. Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:

<math>\begin{align}

                 &&                           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{align}</math>

ExamplesEdit

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.

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

Category of groupsEdit

If Template:Nowrap and Template:Nowrap are group homomorphisms, then so is Template:Nowrap. 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 groupsEdit

If G and H are abelian (i.e., commutative) groups, then the set Template:Nowrap of all group homomorphisms from G to H is itself an abelian group: the sum Template:Nowrap 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 Template:Nowrap is again a group homomorphism.

The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in Template:Nowrap, h, k are elements of Template:Nowrap, and g is in Template:Nowrap, then

Template:Nowrap and Template:Nowrap.

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.

ReferencesEdit

External linksEdit

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