Group homomorphism

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In mathematics, given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : GH 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 element eG 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(x) may be 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(x) 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 (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.

Contents

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

and the image of h to be

The kernel is a normal subgroup of G (in fact, h(g-1 u g) = h(g)-1 h(u) h(g) = h(g)-1 eH h(g) = h(g)-1 h(g) = eH) and the image is a subgroup of H. The homomorphism h is injective (and called a group monomorphism) if and only if ker(h) = {eG}.

The kernel and image h(G) = {h(g), gG} 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.

Examples

  • Consider the cyclic group Z/3Z = {0, 1, 2} and the group of integers Z with addition. The map h : ZZ/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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