Normalizer and centralizer commute with inner automorphisms

Prove that if G is a group, S \subseteq G, and g \in G, then g N_G(S) g^{-1} = N_G(gSg^{-1}) and g C_G(S) g^{-1} = C_G(gSg^{-1}).


Note that

  • x \in gN_G(S)g^{-1}
  • if and only if g^{-1}xg \in N_G(S)
  • if and only if (g^{-1}xg)S(g^{-1}xg) = S
  • if and only if x(gSg^{-1})x^{-1} = gSg^{-1}
  • if and only if x \in N_G(gSg^{-1}),

and that

  • x \in g C_G(S) g^{-1}
  • if and only if g^{-1}xg \in C_G(S)
  • if and only if (g^{-1}xg)s(g^{-1}xg) = s for all s \in S
  • if and only if x(gsg^{-1})x^{-1} = gsg^{-1} for all s \in S
  • if and only if x \in C_G(gSg^{-1}).
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