nLab
centralizer

Contents

Contents

Definition

Definition

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

C G(S){gG|sS(gs=sg)}G 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 cGc \in G which commute with the elements of SS.

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

Properties

Proposition

Given a subset SGS \subset G of a group GG, the centralizer subgroup of SS (Def. ) is a subgroup of the normalizer subgroup:

C G(S)N G(S). C_G(S) \; \subset \; N_G(S) \,.

Proof

Since an element gGg \in G which fixes each element sSs \in S separately already fixes the entire subset as such:

sS(gs=sg)(gS=Sg). \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 T_1 -groups are closed)
If GG is a T 1 T_1 -topological group, then all its centralizer subgroups are closed subgroups.

Proof

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

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

(the adjoint action of GG on itself).

Noticing here that:

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

  2. {s}G\{s\} \subset G is a closed subset, by the assumption that GG is a T 1 T_1 -space (by this Prop.)

it follows that C G({s})GC_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 SS, its centralizer is clearly the intersection of the centralizers of (the singleton sets on) its elements:

C G(S)=sSC G({s}). 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.

References

See also

Last revised on September 4, 2021 at 14:05:43. See the history of this page for a list of all contributions to it.