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

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,

and it also maps inverses to inverses in the sense that

Hence one can say that h "is compatible with the group structure".

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.

Let ${\displaystyle e_{H}}$ be the identity element of the (H, ·) group and ${\displaystyle u\in G}$, then

Now by multiplying for the inverse of ${\displaystyle h(u)}$ (or applying the cancellation rule) we obtain