Group homomorphism

Image of a group homomorphism (h) from G (left) to H (right). The smaller oval inside H is the image of h. N is the kernel of h and aN is a coset of N.

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

$h(u*v)=h(u)\cdot h(v)$

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_H of H, and it also maps inverses to inverses in the sense that

$h\left(u^{-1}\right)=h(u{)}^{-1}.$

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.