# Centerless 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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## Definition

### Symbol-free definition

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

- The center of the group is trivial.
- The natural homomorphism from the group to its automorphism group that sends each element to the conjugation it induces, is injective (viz no two elements induce the same inner automorphism)

### Definition with symbols

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

- The center is the trivial group.
- The natural homomorphism given by , where , is an
*injective*map from to .

## 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.