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

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 *e _{G}* of

*G*to the identity element

*e*of

_{H}*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 *x*_{h}, 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.

Read more about Group Homomorphism: Intuition, Image and Kernel, Examples, The Category of Groups, Types of Homomorphic Maps, Homomorphisms of Abelian Groups

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