Normaliser

Definition

Let $G$ be a group and let $S$ be a subset of $G$. The normaliser of $S$ in $G$ is the set $N_G(S) = \{g \in G : gSg^{-1} = S\}$, where $gSg^{-1} = \{gsg^{-1} : s \in S\}$.

Remarks

The normaliser $N_G(S)$ is a subgroup of $G$, and it contains the centraliser $C_G(S)$. If $H$ is a subgroup of $G$, then $H$ is a normal subgroup of $N_G(H)$. Also, $N_G(H)$ is the largest subgroup of $G$ with this property. Thus $H$ is normal in $G$ exactly when $N_G(H) = G$. The number of conjugates of $H$ in $G$ is the index $[G : N_G(H)]$.

Explanations

L1 human

The normaliser of a subgroup is the set of elements that do not move the subgroup when they conjugate it. They may move the elements of the subgroup around, but the subgroup as a set stays in place. The normaliser is big when the subgroup is close to normal.