This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
A group is said to be capable if it satisfies the following equivalent conditions:
- It is isomorphic to the inner automorphism group of some group. In other words, there is a group such that is isomorphic to the quotient group where is the center of the group.
- Its epicenter is the trivial group.
In terms of the image operator
- The trivial group is capable; it occurs as the inner automorphism group of any abelian group
- A nontrivial cyclic group cannot be capable. This is because there cannot be an element outside the center of a group, which, along with the center, generates the whole group. For full proof, refer: cyclic and capable implies trivial
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|centerless group||center is trivial||A centerless group is isomorphic to its own inner automorphism group.||The [Klein four-group]] is capable but not centerless.|||FULL LIST, MORE INFO|
|simple non-abelian group||non-abelian and simple: no proper nontrivial normal subgroup||(via centerless)||(via centerless)||Centerless group|FULL LIST, MORE INFO|
|almost simple group||between a simple non-abelian group and its automorphism group||(via centerless)||(via centerless)||Centerless group|FULL LIST, MORE INFO|
|characteristically simple non-abelian group||non-abelian and characteristically simple: no proper nontrivial characteristic subgroup||(via centerless)||(via centerless)||Centerless group|FULL LIST, MORE INFO|
|complete group||centerless and every automorphism is inner||(via centerless)||(via centerless)||Centerless group|FULL LIST, MORE INFO|