Isomorphic groups

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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.