nLab centralizer

Definition

Definition

Given a group $G$ and a subset $S \,\subset\, G$ of its underlying set, the centralizer subgroup (also: the commutant) of $S$ in $G$ is the subgroup

$$C_G(S) \;\coloneqq\; \big\{ g \in G \,\vert\, \underset{s \in S}{\forall} ( g \cdot s \,=\, s \cdot g ) \big\} \;\subset\; G$$

of all elements $c \in G$ which commute with the elements of $S$.

Equivalently, the centralizer is the joint fixed point subgroup of the inner automorphisms on $G$ given by conjugation with the elements $s \in S$.

Notice the similarity with but the difference to the concept of normalizer subgroup, cf. Prop. .

Properties

Proposition

Given a subset $S \subset G$ of a group $G$, the centralizer subgroup of $S$ (Def. ) is a subgroup of the normalizer subgroup:

$$C_G(S) \; \subset \; N_G(S) \,.$$

Proof

Since an element $g \in G$ which fixes each element $s \in S$ separately already fixes the entire subset as such:

$$\underset{s \in S}{\forall} \big( g \cdot s \,=\, s \cdot g \big) \;\;\;\;\; \Rightarrow \;\;\;\;\; \big( g \cdot S \,=\, S \cdot g \big) \,.$$

Proposition

(centralizers in $T_1$-groups are closed)
If $G$ is a $T_1$-topological group, then all its centralizer subgroups are closed subgroups.

Proof

First consider a singleton set $S = \{s\}$. By definition, the centralizer of a single element $s \in G$ is the preimage of itself under the function

$$\array{ G &\xrightarrow{\;\;}& G \\ g &\mapsto& g \cdot s \cdot g^{-1} \,. }$$

(the adjoint action of $G$ on itself).

Noticing here that:

1. this is continuous function, by the axioms on a topological group;

2. $\{s\} \subset G$ is a closed subset, by the assumption that $G$ is a $T_1$-space (by this Prop.)

it follows that $C_G(\{s\}) \subset G$ is the continuous preimage of a closed subset and hence is itself closed (by this Prop.).

Now for a general set $S$, its centralizer is clearly the intersection of the centralizers of (the singleton sets on) its elements:

$$C_G(S) \;=\; \underset{ s \in S }{\cap} C_G\big(\{s\}\big) \,.$$

Since each of the factors on the right isclosed, by the previous argument, the general centralizer subgroup is an intersection of closed subsets and hence itself a closed subset.

Related concepts

• stabilizer subgroup

• normalizer subgroup

References

See also

• Wikipedia, Centralizer and normalizer