P-subset

Lemma 2.3.9 Letp be a prime number, and letT be a closed p-subset ofS. Then there exists an element t in T 1 such that 1 = nt. 

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). 

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. 

Corollary 5.6.8 Let p be a prime number. Then T is a closed p-subset of S if and only if each composition factor of T is thin and has valency p. 

Proof. Assume first that T is a closed p-subset of S. Then, T is p-valenced and nr is a power of p. Thus, by Theorem 5.6.7, T is residually thin. Thus, by Theorem 5.6.1, each composition factor of S is thin. 

Conversely, let us assume that each composition factor of T is thin and has valency p. We may assume that 1 / T. Then T possesses a thin closed subset U of valency p. Thus, by induction, T//U is a closed p-subset of S//U. Thus, by Corollary 4.3.2(i), T is a closed p-subset of S. 

Corollary 5.6.9 Letp be a prime number, assume thatT is a closed p-subset of S, and let U be a closed p-subset of S such that T C Ns(U). Then TU is a closed p-subset of S. 

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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