# C-closed self-centralizing subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: c-closed subgroup and self-centralizing subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed a **c-closed self-centralizing subgroup** or a **centralizer of Abelian subgroup** if it satisfies the following equivalent conditions:

- Its centralizer equals its center and it equals the centralizer of its center (all relative to the whole group).
- It is self-centralizing (i.e., it contains its own centralizer in the whole group) and also occurs as the centralizer of some subgroup of the whole group.
- It occurs as the centralizer of some Abelian subgroup of the whole group.

### Definition with symbols

A subgroup of a group is termed a **c-closed self-centralizing subgroup** or a **centralizer of Abelian subgroup** if it satisfies the following equivalent conditions:

- and .
- and there exists a subgroup such that .
- There exists an Abelian subgroup of such that .