# Group isomorphism

### From Online Dictionary of Crystallography

Isomorphisme de groupes (Fr). Gruppenisomorphismus (Ge). Isomorfismo fra gruppi (It). 同形 (Ja). Isomorfismo de grupos (Sp).

A group isomorphism is a special type of group homomorphism. It is a mapping between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the respective group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. Isomorphic groups have the same properties and the same structure of their multiplication table.

Let (G, *) and (H, #) be two groups, where '*' and '#' are the binary operations in G and H, respectively. A group isomorphism from (G, *) to (H, #) is a bijection from G to H, i.e. a bijective mapping f : GH such that for all u and v in G one has

f (u * v) = f (u) # f (v).

Two groups (G, *) and (H, #) are isomorphic if an isomorphism between them exists. This is written:

(G, *) [math]\cong[/math] (H, #).

If H = G and the binary operations # and * coincide, the bijection is an automorphism.