nLab
centralizer

Contents

Definition

The centralizer subgroup (also: commutant) of a subset $S$ in (the set underlying) a group $G$ is the subgroup

C_G(S) \subset G

of all elements $c \in G$ which commute with $S$ , hence such that $c \cdot s = s \cdot c$ for all $s \in S$ .

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

The centralizer of $S$ is clearly a subgroup of its normalizer

C_G(S) \subset N_G(S)

as fixing the set $g H = H g$ is a weaker requirement than $g h=h g$ for all $h\in H$ .