# Capable group

From Groupprops

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 propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

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 group property of being a capable group is obtained by applying the image operator for the quotient-defining function sending each group to its inner automorphism group.

## Facts

- 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

### Stronger 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 |