group homomorphism#Types

{{Short description|Mathematical function between groups that preserves multiplication structure}}

Image:Group homomorphism ver.2.svg of h. N is the kernel of h and aN is a coset of N.]]

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{{anchor|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 $u, g$ :

: $\begin{align}
h\left(g^{-1} \circ u \circ g\right) &= h(g)^{-1} \cdot h(u) \cdot h(g) \\
&= h(g)^{-1} \cdot e_H \cdot h(g) \\
&= h(g)^{-1} \cdot h(g) = e_H,
\end{align}$

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 {{nowrap|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:

: $\begin{align}
&& h(g_1) &= h(g_2) \\
\Leftrightarrow && h(g_1) \cdot h(g_2)^{-1} &= e_H \\
\Leftrightarrow && h\left(g_1 \circ g_2^{-1}\right) &= e_H,\ \operatorname{ker}(h) = \{e_G\} \\
\Rightarrow && g_1 \circ g_2^{-1} &= e_G \\
\Leftrightarrow && g_1 &= g_2
\end{align}$

Examples

Consider the cyclic group Z{{sub|3}} = (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.

{{bulleted list|

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 $\Phi: (\mathbb{Z}, +) \rightarrow (\mathbb{R}, +)$ , 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 {{nowrap|h : G → H}} and {{nowrap|k : H → K}} are group homomorphisms, then so is {{nowrap|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 {{nowrap|Hom(G, H)}} of all group homomorphisms from G to H is itself an abelian group: the sum {{nowrap|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 {{nowrap|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 {{nowrap|Hom(K, G)}}, h, k are elements of {{nowrap|Hom(G, H)}}, and g is in {{nowrap|Hom(H, L)}}, then

:{{nowrap|1=(h + k) ∘ f = (h ∘ f) + (k ∘ f)}} and {{nowrap|1=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

{{cite book

| last1 = Dummit

| first1 = D. S.

| last2 = Foote

| first2 = R.

| title = Abstract Algebra

| publisher = Wiley

| pages = 71–72

| year = 2004

| edition = 3rd

| isbn = 978-0-471-43334-7

}}

{{Lang Algebra|edition=3r}}

External links

{{MathWorld|title=Group Homomorphism|urlname=GroupHomomorphism|author=Rowland, Todd|author2=Weisstein, Eric W.|name-list-style=amp}}

Category:Group theory

Category:Morphisms