Normalizer

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, $A \subseteq N(A)$.

When $A$ is a subgroup of $G$, $N(A)$ is the largest subgroup of $G$ of which $A$ is a normal subgroup.

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