Constrained Sets of Involutions

Each element in T is assumed to have valency at most 2. Thus, as 17 C T, each element in U has valency at most 2. Thus, we obtain from Lemma 6.3.5 that, for any two elements w in xU and u in U, acts transitively on (the two elements of) wu. Thus, we just have to show that G acts transitively on xU cf. Theorem 6.3.1 and Theorem 6.3.3. 

In order to show this latter condition we fix elements y and 2 in xU. According to Lemma 6.3.4, we may assume that there exists an element f in T such that 2 e yt t. Since 2 e yt t, there exists an element w in yt such that 2 e wt. 

In accordance with Section 3.4 we shall write, for each element s in (L), S i(s) instead of S i(s, L). 

Throughout this section, the letter L stands for a constrained set of involutions. 

Let us denote by s the uniquely determined element in S which satisfies 2 G xs. We have to prove that 2% xys. 

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In the last two sections of this chapter, we impose specific conditions on sets of involutions. In Section 3.5, we look at constrained sets of involutions, and in the last of the six sections of this chapter, we look at constrained sets of involutions which satisfy the exchange condition. According to what we said in the preface of this monograph, such sets of involutions are called Coxeter sets. 

In the last of the six sections of this chapter, we shall briefly look at scheme rings of S under the assumption that S is generated by a constrained set of involutions. 

Closed subsets generated by sets of involutions turn out to be an interesting subject, especially if one imposes appropriate extra conditions on the set of the generating involutions. As an example, we introduce constrained sets of involutions as another example, we look at sets of involutions satisfying the exchange condition. A constrained set of involutions which satisfies the exchange condition will be called a Coxeter set. 

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