Normalizer

From AoPSWiki

A normalizer is a part of a group.

Let $A$ be a subset of a group $G$ . An element $b$ of $G$ is said to normalize $A$ if $bAb^{-1} = A$ . A subset $B$ of $G$ is said to normalize $A$ if all its elements normalize $A$ . The set of all elements of $G$ that normalize $A$ is called the normalizer of $A$ . It is often denoted as $N_G(A)$ , or $N(A)$ , when there is no risk of confusion. It is evidently a subgroup of $G$ ; for $e \in N(A)$ ; if $b,c$ normalize $A$ , then $(bc)A(bc)^{-1} = bcAc^{-1}b^{-1} = bAb^{-1} = A;$ and if $bAb^{-1} = A$ , then $A = b^{-1}Ab$ . Evidently, $A \subseteq N(A)$ .