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 $E$ .

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Centralizer

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