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
of all elements $c \in G$ which commute with the elements of $S$.
Notice the similarity with but the difference to the concept of normalizer subgroup, see Prop. .
Given a subset $S \subset G$ of a group $G$, the centralizer subgroup of $S$ (Def. ) is a subgroup of the normalizer subgroup:
Since an element $g \in G$ which fixes each element $s \in S$ separately already fixes the entire subset as such:
(centralizers in $T_1$-groups are closed)
If $G$ is a $T_1$-topological group, then all its centralizer subgroups are closed subgroups.
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
(the adjoint action of $G$ on itself).
Noticing here that:
this is continuous function, by the axioms on a topological group;
$\{s\} \subset G$ 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:
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.
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.