# Centerless group

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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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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## Definition

### Symbol-free definition

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

### Definition with symbols

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

• The center $Z(G)$ is the trivial group.
• The natural homomorphism $G \to \operatorname{Aut}(G)$ given by $g \mapsto c_g$, where $c_g = x \mapsto gxg^{-1}$, is an injective map from $G$ to $\operatorname{Aut}(G)$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
simple non-abelian group non-abelian and simple: has no proper nontrivial normal subgroup simple and non-abelian implies centerless symmetric group:S3 is a centerless group that is not simple. |FULL LIST, MORE INFO
almost simple group in between a simple non-abelian group and its automorphism group almost simple implies centerless symmetric group:S3 is a centerless group that is not almost simple. |FULL LIST, MORE INFO
complete group centerless and every automorphism is inner (by definition) alternating group:A4 is centerless but not complete, it has outer automorphisms realized as conjugations via its embedding as A4 in S4. |FULL LIST, MORE INFO
characteristically simple non-abelian group non-abelian and characteristically simple: has no proper nontrivial characteristic subgroup characteristically simple and non-abelian implies centerless symmetric group:S3 is a centerless group that is not simple. |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
capable group can be expressed as (up to isomorphism) the inner automorphism group of some group; equivalently, the epicenter is trivial. centerless implies capable -- basically, it is isomorphic to its own inner automorphism group |FULL LIST, MORE INFO
stem group the center is contained in the derived subgroup. |FULL LIST, MORE INFO
unicentral group epicenter equals center. |FULL LIST, MORE INFO

## Metaproperties

### Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A direct product of centerless groups is centerless. This follows from the general fact that the center of a direct product equals the direct product of the individual centers.