## A criterion for finite subgroup normalcy, on generating sets of the subgroup and group

Let $G$ be a group and $N \leq G$ a finite subgroup, such that $G = \langle T \rangle$ and $N = \langle S \rangle$. Prove that $N$ is normal in $G$ if and only if $tSt^{-1} \subseteq N$ for all $t \in T$.

The $(\Rightarrow)$ direction is clear. $(\Leftarrow)$ Suppose $tSt^{-1} \subseteq N$ for all $t \in T$; then $tNt^{-1} = t\langle S \rangle t^{-1} = \langle tSt^{-1} \rangle$ $\subseteq N$ for all $t$. Since $N$ is finite, $tNt^{-1} = N$ for all $t \in T$. So $T \subseteq N_G(N)$, and we have $G = \langle T \rangle \subseteq N_G(N)$. Thus $N$ is normal in $G$.

A little simplication: Suppose that $tSt^{-1} \subseteq N$ for all $t \in T$. Then $\langle tSt^{-1} \rangle = tNt^{-1} \subseteq N$. Since $N$ is finite, this means that $t \in N_G (N)$ for all $t \in T$. So $\langle t | t \in T \rangle = G \subseteq N_G(N)$.