# Centralizer-connected 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 **centralizer-connected** if the centralizer of any element is either the whole group or has infinite index in the group., or equivalently, if every conjugacy class with more than one element has infinite size.

### Reason for the name

The concept of **centralizer-connected** is closely related to the concept of connected. In particular, for a connected topological group, every conjugacy class is connected (as it arises as the image of the group under a continuous map). Hence, it must either be a single point or infinite in size, so it must be centralizer-connecteed.

## Relation with other properties

### Stronger properties

These are stronger properties that are satisfied by groups with some additional structure:

## Facts

In a centralizer-connected group, any finite normal subgroup is central.