Contents

group theory

# 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 $h\in H$.