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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