Definition:Centralizer/Group Element

Definition
Let $\left({G, \circ}\right)$ be a group.

Let $a \in \left({G, \circ}\right)$.

The centralizer of $a$ (in $G$) is defined as:


 * $C_G \left({a}\right) = \left\{{x \in G: x \circ a = a \circ x}\right\}$

That is, the centralizer of $a$ is the set of elements of $G$ which commute with $a$.

Also known as
Some sources call this the normalizer of $a$ in $G$ but that term generally has another meaning.

The UK English spelling of this is centraliser.

Also see

 * Stabilizer of Element under Conjugacy Action is Centralizer