In mathematics, given two groups, and, a group homomorphism from to is a functionh : G → H such that for all u and v in G it holds that 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 elementeG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure". Older notations for the homomorphism h may be xh or xh, 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 their arguments, omitting brackets, so that h 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 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 ⋅ h = h. In other words, the group H in some sense has a similar algebraic structure as G and the homomorphism h preserves that.
Types of group homomorphism
;Monomorphism: A group homomorphism that is injective ; i.e., preserves distinctness. ;Epimorphism: A group homomorphism that is surjective ; 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 homomorphism, h: G → G; the domain and codomain are the same. Also called an endomorphism of G. ;Automorphism: An endomorphism that is bijective, and hence an isomorphism. 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. As an example, the automorphism group of contains only two elements, the identity transformation and multiplication with −1; it is isomorphic to Z/2Z.
We define the kernel of h to be the set of elements in G which are mapped to the identity in H and the image of h to be 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 is isomorphic to the quotient groupG/ker h. The kernel of h is a normal subgroup of G and the image of h is a subgroup of H: If and only if, the homomorphism, h, is a group monomorphism; i.e., h is injective. Injection directly gives that there is a unique element in the kernel, and a unique element in the kernel gives injection:
Examples
Consider the cyclic groupZ/3Z = and the group of integers Z with addition. The map h : Z → Z/3Z with h = umod 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 numbersR with addition to the group of non-zero real numbers R* with multiplication. The kernel is 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, 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.
If G and H are abelian groups, then the set of all group homomorphisms from G to H is itself an abelian group: the sum of two homomorphisms is defined by The commutativity of H is needed to prove that is again a group homomorphism. The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in, h, k are elements of, and g is in, then Since the composition is associative, this shows that the set End 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-mmatrices 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.
