Tour:Isomorphic groups
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This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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WHAT YOU NEED TO DO:
- Understand the definition of isomorphic groups
- Convince yourself of why being isomorphic is an equivalence relation.
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.
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