# Isomorphic groups

## Contents

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This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

## Definition

Two groups  and  are termed isomorphic groups, in symbols  or , if there exists an isomorphism of groups from  to .

The relation of being isomorphic is an equivalence relation on groups:

• Reflexivity: The identity map is an isomorphism from any group to itself.
• Symmetry: The inverse of an isomorphism is an isomorphism.
• Transitivity: if  is isomorphic to  and  is isomorphic to , then  is isomorphic to , via the isomorphism obtained by composing the isomorphisms from  to  and from  to .

As far as the group structure is concerned, isomorphic groups behave in exactly the same way, so constructions and properties for groups are all studied upto isomorphism-invariance.

## Relation with other relations

All equivalence relations and symmetric relations of groups usually studied are weaker than the relation of being isomorphic. For a list, see Category:Equivalence relations on groups.