# Group isomorphism

### From Online Dictionary of Crystallography

Isomorphisme entre groupes (*Fr*). Gruppenisomorphismus (*Ge*). Isomorfismo fra gruppi (*It*). 同形 (*Ja*).

A **group isomorphism** is a special type of group homomorphism. It is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given 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 respective binary operations in *G* and in *H*. A *group isomorphism* from (*G*, *) to (*H*, #) is a bijection from *G* to *H*, *i*.*e*. a bijective function *f* : *G* → *H* such that for all *u* and *v* in *G* it holds that

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

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

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

If *H* = *G* and # = * then the bijection is an automorphism.