Centralizer

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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 $G$ .

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Centralizer

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Centralizers in Groups

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