# Difference between revisions of "Group isomorphism"

### From Online Dictionary of Crystallography

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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. | 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 | + | Let (''G'', *) and (''H'', #) be two groups, where "*" and "#" are the respective [[binary operation]]s 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 |

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## Revision as of 18:32, 21 December 2008

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.