Isomorphic groups
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Isomorphic groups, all facts related to Isomorphic groups) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |[SHOW MORE]
This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.
Contents
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.
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.