# Centralizer

A centralizer is part of an algebraic structure.

Specifically, let $E$ be a magma, and let $X$ be a subset of $E$. The centralizer $X'$ of $X$ is the set of elements of $E$ which commute with every element of $X'$.

If $X \subseteq Y$ are subsets of a magma $E$, then $Y' \subseteq X'$. The bicentralizer $X''$ of $X$ is the centralizer of $X'$. Evidently, $X \subseteq X''$. The centralizer of the bicentralizer, $X'''$, is equal to $X'$, for $X' \subseteq X'''$, but $X \subseteq X''$, so $X''' \subseteq X'$.

If the magma $E$ is associative, then the centralizer of $X$ is also the centralizer of the subset of $E$ genererated by $X$, and the centralizer of $X$ is furthermore an associative sub-magma of $E$. If $E$ is a group, then the centralizer of $X$ is a subgroup, though not necessarily normal. The centralizer of $E$ is also called the center of $E$.

## Centralizers in Groups

If $G$ is a group, then an element $b$ of $G$ is said to centralize $A$ if it commutes with every element of $A$; that is, if $bab^{-1} = a$ for all $a \in A$. A subset $B$ of $G$ is said to centralize $A$ if all its elements centralize $A$. The centralizer of $A$, denoted $C_G(A)$, or $C(A)$ when there is no risk of confusion, is the set of elements that centralize $A$. It is evidently a subgroup of $G$.