Isomorphic 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.