Group homomorphism
Algebraic structure → Group theory Group theory 

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
 [math]\displaystyle{ h(u*v) = h(u) \cdot h(v) }[/math]
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,
 [math]\displaystyle{ h(e_G) = e_H }[/math]
and it also maps inverses to inverses in the sense that
 [math]\displaystyle{ h\left(u^{1}\right) = h(u)^{1}. \, }[/math]
Hence one can say that h "is compatible with the group structure".
Older notations for the homomorphism h(x) may be x^{h} or x_{h}, though this may be confused as an index or a general subscript. In automata theory, sometimes homomorphisms are written to the right of their arguments without parentheses, so that h(x) becomes simply [math]\displaystyle{ xh }[/math].
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 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.
Types
 Monomorphism
 A group homomorphism that is injective (or, onetoone); 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 and are identical for all practical purposes.
 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
 [math]\displaystyle{ \operatorname{ker}(h) := \left\{u \in G\colon h(u) = e_{H}\right\}. }[/math]
and the image of h to be
 [math]\displaystyle{ \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 and the image of h is a subgroup of H:
 [math]\displaystyle{ \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]
If and only if ker(h) = {e_{G}}, the homomorphism, h, is a group monomorphism; i.e., h is injective (onetoone). Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:
 [math]\displaystyle{ \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]
Examples
 Consider the cyclic group Z_{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.

 [math]\displaystyle{ G \equiv \left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \bigg a \gt 0, b \in \mathbf{R}\right\} }[/math]
For any complex number u the function f_{u} : G → C^{*} defined by:
 [math]\displaystyle{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \mapsto a^u }[/math]
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:
 [math]\displaystyle{ f_u(a) = a^u }[/math]
 The exponential map yields a group homomorphism from the group of real numbers R with addition to the group of nonzero 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 nonzero 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.
Category of groups
If h : G → H and k : H → K are group homomorphisms, then so is k ∘ h : G → K. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.
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) ∘ f = (h ∘ f) + (k ∘ f) and 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 mbym 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 wellbehaved kernels makes this category the prototypical example of an abelian category.
See also
References
 Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3rd ed.). Wiley. pp. 71–72. ISBN 9780471433347.
 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: SpringerVerlag, ISBN 9780387953854
External links
 Rowland, Todd. "Group Homomorphism". http://mathworld.wolfram.com/GroupHomomorphism.html.
Original source: https://en.wikipedia.org/wiki/Group homomorphism.
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