Basic properties of normalizers with respect to a subgroup

Let G be a group, H \leq G, and A \subseteq G. Define N_H(A) = \{ h \in H \ |\ hAh^{-1} = A \}. Show that N_H(A) = N_G(A) \cap H and deduce that N_H(A) \leq H.


(\subseteq) That N_H(A) \subseteq N_G(A) is clear, as is N_H(A) \subseteq H. Thus N_H(A) \subseteq N_G(A) \cap H. (\supseteq) Suppose h \in N_G(A) \cap H. Then we have h \in H and hAh^{-1} = A, so that h \in N_H(A).

Now N_H(A) \subseteq H. By a lemma to a previous problem, then, N_H(A) \leq H.

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