Sylow Subsets

In this section, S is assumed to have finite valency. We fix a prime number and call it p. 

The following lemma on finite groups is commonly referred to Augustin-Louis 

Lemma 4.5.1 Let T be a thin closed subset of S such that p divides n. Then T contains a closed subset of valency p. 

Assume T to be a minimal counterexample. Then T does not contain closed subsets different from T the valency of which is divisible by p. 

Let T be a closed subset of S. Note that a p -subset of T is the same as a p-subset of T. We also speak about Sylow p-subsets of T instead of Hall p -subsets of T.1 Our notation for the set of all Sylow p-subsets of T will be SyiP(T). 

Let [/ be a closed subsets of T such that 1 U T. Then, by Lemma 4.3.3(i), p divides nTHu- Thus, as 1 U, the minimal choice of T yields a closed subset P of S such that U CV and nyjju = p. Let t be an element in V U. Then (t) has valencyp. 

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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. 

Sylow p-subsets of a closed subset T of S are particularly interesting if T is p-valenced. 

Theorem 4.5.3 Each p-valenced closed subset of S possesses at least one Sylow p-subset. 

Proof. Let T be a p-valenced closed subset of S. Then, Proposition 4.5.2 says that, for each power q of p which divides the valency of T, T possesses a closed p-subset of valency q. In particular, T possesses a Sylow p-subset. 

According to Theorem 4.5.3, each p-valenced closed subset of S possesses at least one Sylow p-subset. Generalizing this theorem we shall now say a little bit more about the number of Sylow p-subsets of such a closed subset of S. 

Theorem 4.5.7 The number of Sylow p-subsets of a p-valenced closed subset of S is congruent to 1 modulo p. 

Quotient schemes are introduced in the fourth chapter of this monograph. Factorization over non-normal closed subsets provides us with a particularly smooth approach to a generalization of Ludwig Sylow s theorems on finite... 

Since we are assuming T to be a Sylow p-subset of P, p does not divide nynT- Thus, as nu//T divides nynT, p does not divide nu//T- Thus, as p divides (nynu — l)riij//T, P divides ny//u — 1. 

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