Capable group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

A group G is said to be capable if it satisfies the following equivalent conditions:

  1. It is isomorphic to the inner automorphism group of some group. In other words, there is a group H such that G is isomorphic to the quotient group H/Z(H) where Z(H) is the center of the group.
  2. 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

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