Conjugates of Closed Subsets

Lemma 2.5.8 Let R be a nonempty subset of S, and let T be a closed subset of S. Then the following hold. 

Lemma 2.5.9 For each closed subset T of S, the following hold. 

Let P and Q be nonempty subsets of S. We call Q a conjugate of P if there exists an element s in S such that Q = s Ps. 

Since q has been chosen arbitrarily in s Ks T)s, we thus have shown that s Ks(T)s C Ks(s Ts). 

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In Section 2.5, we define the normalizer and the strong normalizer of closed subsets. In the last of the six sections of this chapter, we introduce conjugates of closed subsets. Conjugates are related to normalizers and strong normalizers and will play a role in Section 4.4 when we investigate Sylow subsets. 

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