# Group isomorphism

### From Online Dictionary of Crystallography

جماعه التماثل (*Ar*). Isomorphisme de groupes (*Fr*). Gruppenisomorphismus (*Ge*). Isomorfismo fra gruppi (*It*). 同形 (*Ja*). Изоморфизм групп (*Ru*). 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* : *G* → *H* 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.