Definition:Centralizer/Group Subset

Definition

Let $\struct {G, \circ}$ be a group.

Let $S \subseteq G$.

The centralizer of $S$ (in $G$) is the set of elements of $G$ which commute with all $s \in S$:

$\map {C_G} S = \set {x \in G: \forall s \in S: x \circ s = s \circ x}$

Also denoted as

The notation $\map C {S; G}$ is sometimes seen for the centralizer of $S$ in $G$.

Linguistic Note

The UK English spelling of centralizer is centraliser.

Sources

• 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Exercise $(11)$