Isomorphic groups

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 $G$ and $H$ are termed isomorphic groups, in symbols $G\cong H$ or $H\cong G$, if there exists an isomorphism of groups from $G$ to $H$.

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 $G$ is isomorphic to $H$ and $H$ is isomorphic to $K$, then $G$ is isomorphic to $K$, via the isomorphism obtained by composing the isomorphisms from $G$ to $H$ and from $H$ to $K$.

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.