Isomorphic groups
From Groupprops
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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.