# Spooky algebra II

Exhibit a group $G$, a set of generators $X$ for $G$ and a subgroup $H$ such that $xHx^{-1} \subseteq H$ for all $x\in X$ and yet $H$ is not normal in $G$.

1. Let $M=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $M=\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ and consider the group $G=\langle M, N\rangle$. Since $NMN^{-1}= M^2$, $\langle M\rangle$ is stable under conjugation by both $M$ and $N$ yet $N^{-1}MN=\begin{pmatrix} 1 & 1/2 \\ 0 & 1 \end{pmatrix}$ and so is not a normal subgroup.