Centraliser

Definition

Let $G$ be a group and let $S$ be a subset of $G$. The centraliser of $S$ in $G$ is the set $C_G(S) = \{g \in G : gs = sg \text{ for all } s \in S\}$. For an element $a \in G$, the notation $C_G(a)$ means $C_G(\{a\})$.

Remarks

The centraliser $C_G(S)$ is a subgroup of $G$. The centraliser of $G$ itself is the centre $Z(G)$. The element $a$ lies in its own centraliser $C_G(a)$, and so does each power of $a$.

Explanations

L1 human

The centraliser of an element is the set of elements that commute with it. If the element commutes with everything, its centraliser is the whole group. The size of the centraliser tells you how many conjugates the element has.