Centralizer

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

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 $S$ of a group $G$, the centralizer of $S$ in $G$, denoted as $C_{G}(S)$, is defined as the subgroup of $G$ comprising all $x$ such that $xg=gx$ for all $g$ in $S$. For any $S$, the centralizer $C_{G}(S)$ is a subgroup of the group $G$. 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:

• $S_{1}\subseteq S_{2}$ implies $C_{G}(S_{2})\subseteq C_{G}(S_{1})$
• $S\subseteq C_{G}(C_{G}(S))$

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.