Group homomorphism

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

Related Topics:
Mathematics - Groups - Function

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: h(u * v) = h(u) · h(v)

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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 h(u-1) = h(u)-1. Hence one can say that h "is compatible with the group structure".

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Older notations for the homomorphism h(x) may be xh, though this may be confused as an index or a general subscript.

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A more recent trend is to write group homomorphisms on the right of their

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arguments, omitting brackets, so that h(x) becomes simply x h.

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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.

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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.

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