Difference between revisions of "Isomorphic groups"

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(Relation with other relations)
 
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==Relation with other relations==
 
==Relation with other relations==
  
===Weaker relations===
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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]].
 
 
{| class="sortable" border="1"
 
! Relation !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 
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| [[Stronger than::1-isomorphic groups]] || || || || {{intermediate notions short|1-isomorphic groups|isomorphic groups}}
 
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| [[Stronger than::directed power graph-equivalent groups]] || || || || {{intermediate notions short|directed power graph-equivalent groups|isomorphic groups}}
 
|-
 
| [[Stronger than::undirected power graph-equivalent groups]] || || || || {{intermediate notions short|undirected power graph-equivalent groups|isomorphic groups}}
 
|}
 

Latest revision as of 14:43, 7 August 2010

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.