Group Isomorphism - Definition and Notation

Definition and Notation

Given two groups (G, *) and (H, ), a group isomorphism from (G, *) to (H, ) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function such that for all u and v in G it holds that

.

The two groups (G, *) and (H, ) are isomorphic if there exists an isomorphism from one to the other. This is written:

Often shorter and simpler notations can be used. When the relevant group operations are unambiguous they are omitted and one writes:

Sometimes one can even simply write G = H. Whether such a notation is possible without confusion or ambiguity depends on context. For example, the equals sign is not very suitable when the groups are both subgroups of the same group. See also the examples.

Conversely, given a group (G, *), a set H, and a bijection, we can make H a group (H, ) by defining

.

If H = G and = * then the bijection is an automorphism (q.v.)

Intuitively, group theorists view two isomorphic groups as follows: For every element g of a group G, there exists an element h of H such that h 'behaves in the same way' as g (operates with other elements of the group in the same way as g). For instance, if g generates G, then so does h. This implies in particular that G and H are in bijective correspondence. Thus, the definition of an isomorphism is quite natural.

An isomorphism of groups may equivalently be defined as an invertible morphism in the category of groups, where invertible here means has a two-sided inverse.