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.

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

MonomorphismTemplate: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 kernel

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

$\ker (h):=\{u\in G:h(u)=e_H\}.$

and the image of h to be

$\mathrm{im}(h):=h(G)\equiv \{h(u):u\in G\}.$

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.

The kernel of h is a normal subgroup of G. Assume $u\in \ker (h)$ and show $g^{-1}\circ u\circ g\in \ker (h)$ for arbitrary $u,g$:

$\begin{array}{cc}h(g^{-1}\circ u\circ g)& =h(g{)}^{-1}\cdot h(u)\cdot h(g)\\ & =h(g{)}^{-1}\cdot e_H\cdot h(g)\\ & =h(g{)}^{-1}\cdot h(g)=e_H,\end{array}$

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:

$\begin{array}{cccc}& & h(g_1)& =h(g_2)\\ \iff & & h(g_1)\cdot h(g_2{)}^{-1}& =e_H\\ \iff & & h(g_1\circ g_2^{-1})& =e_H,\ker (h)=\{e_G\}\\ \Rightarrow & & g_1\circ g_2^{-1}& =e_G\\ \iff & & g_1& =g_2\end{array}$

Examples

• Consider the cyclic group ZTemplate:Sub = (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 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 $\mathrm{\Phi }:(\mathbb{N},+)\to (ℝ,+)$, defined by $\mathrm{\Phi }(x)=\sqrt[]{2}x$ is a homomorphism.
• Consider the two groups $({ℝ}^{+},*)$ and $(ℝ,+)$, represented respectively by $G$ and $H$, where ${ℝ}^{+}$ is the positive real numbers. Then, the function $f:G\to H$ defined by the logarithm function is a homomorphism.

Category of groups

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 groups

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.

See also

• Homomorphism
• Fundamental theorem on homomorphisms
• Quasimorphism
• Ring homomorphism

References

• Template:Cite book
• Template:Lang Algebra

External links

• Template:MathWorld