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.

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Comments

  • Gobi Ree  On November 21, 2011 at 11:49 pm

    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).

    • nbloomf  On November 22, 2011 at 10:11 am

      That is a vast improvement. Thanks!

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