## A criterion for subgroup normalcy

1. Let $G$ be a group and let $N \leq G$. Prove that $N$ is normal in $G$ if and only if $gNg^{-1} \subseteq N$ for all $g \in G$.
2. Let $G = GL_2(\mathbb{Q})$, let $N$ be the subgroup of upper triangular matrices with integer entries and 1s on the diagonal, and let $g$ be the diagonal matrix with entries 2, 1. Show that $gNg^{-1} \subseteq N$ but that $g$ does not normalize $N$.

1. (1) Suppose first that $N$ is normal. Then $N_G(N) = G$; thus, for all $g \in G$, we have $gNg^{-1} = N$. In particular, for all $g \in G$, $gNg^{-1} \subseteq N$. (2) Suppose $gNg^{-1} \subseteq N$ for all $g \in G$. Then for all $g \in G$, we have $N = gg^{-1}Ngg^{-1} \subseteq gNg^{-1}$, so that $gNg^{-1} = N$. Hence $N_G(N) = G$.
2. Let $a = \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix} \in N$. Clearly then $gag^{-1} = \begin{bmatrix} 1 & 2a \\ 0 & 1 \end{bmatrix} \in N$, so that $gNg^{-1} \subseteq N$. $g$ does not normalize $N$, however, because no matrix in $N$ whose (1,2) entry is odd is the $g$-conjugate of any element in $N$.
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