# Centralizer

*For centralizer as a subgroup property, refer c-closed subgroup*

## Contents

## Definition

### Symbol-free definition

Given any subset of a group, the **centralizer** (**centraliser** in British English) of the subset is defined as the set of all elements of the group that commute with *every* element in the subset. Clearly, the centralizer of any subset is a subgroup. The centralizer of any subset of a group is a subgroup of the group.

### Definition with symbols

Given any subset of a group , the centralizer of in , denoted as , is defined as the subgroup of comprising all such that for all in . For any , the centralizer is a subgroup of the group . `For full proof, refer: Centralizer of subset of group is subgroup`

## Facts

### Order of the centralizer of a single permutation

`Further information: conjugacy class size formula for symmetric group`

## As a Galois correspondence

### Brief description

The centralizer operator can be viewed as a Galois correspondence from the collection of subsets of the group to itself. That is, it satisfies the following two properties:

- implies

This essentially follows because the centralizer map arises as the Galois correspondence corresponding to the symmetric relation of commutation between elements of the group.

### Implications

The implication of the above Galois correspondence is as follows. Define the bicentralizer of a subset as the centralizer of its centralizer. Then, a subset equals its own bicentralizer if and only if it occurs as a centralizer of some subset. Such a subset is termed a c-closed subgroup. In particular, it is a subgroup. Also, the centralizer of any subset equals the centralizer of the subgroup it generates.

## Relation between a subgroup and its centralizer

### Subgroups contained in their centralizer

A subgroup of a group is contained in its centralizer if and only if, as an abstract group, the subgroup is an Abelian group.

### Subgroups containing their centralizers

A subgroup of a group that contains its own centralizer is termed a self-centralizing subgroup.

### Subgroups whose centralizer is the whole group

The centralizer of a subgroup is the whole group if and only if the subgroup is a central subgroup, viz it is contained in the center of the whole group.

### Subgroups whose centralizer completes them

A subgroup whose product with its centralizer is the whole group is termed a central factor.

## Computation

`Further information: Centralizer-finding problem`

The problem of finding the centralizer of a single element (or equivalently of a cyclic subgroup) is polynomial-time equivalent to the set stabilizer problem. The idea is to view it as a partition stabilizer problem.