Tour:Isomorphic groups

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This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Isomorphism of groups| UP: Introduction four (beginners)| NEXT: Group of integers
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part
WHAT YOU NEED TO DO:
  • Understand the definition of isomorphic groups
  • Convince yourself of why being isomorphic is an equivalence relation.

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.

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour). If you found anything difficult or unclear, make a note of it; it is likely to be resolved by the end of the tour.
PREVIOUS: Isomorphism of groups | UP: Introduction four (beginners) | NEXT: Cyclic group