Schur group

Let x be an element in X, and let T be a closed subset of S. It is easy to see that the set of all bijective faithful maps from xT to xT is a group with respect to composition. We call this group the Schur group of T with respect to x. If T = S, we just speak about the Schur group of S. 

The Schur group is the subject of the third section of this chapter. We shall see, in particular, that S is schurian if and only if S is isomorphic to a quotient scheme of a thin scheme, and this is the case if and only if, modulo the group... 

In Section 6.4, we use previously obtained results about Schur groups in order to establish a recognition theorem for certain schemes of finite valency all elements of which have valency at most 2. The theorem is related to one of the most significant results in finite group theory, to George Glauberman s Z -Theorem. 

Theorem 6.3.3 A closed subset T of S is schurian if and only if for each element x in X, the Schur group G of T with respect to x acts transitively on xT and, for each element t in T, Gx acts transitively on xt. 

In Lemma 6.3.5 and Lemma 6.3.6 we investigated the Schur group of closed subsets of S in which each element has valency at most 2. From Lemma 1.5.6(ii) we know that, for each element s in S with ns = 2, us s G 2,3. In this section, we focus on closed subsets T of S in which each element t satisfies nt <2 and nt t 2. 

We shall first prove that a closed subset T of S is schurian if each element t of T satisfies nt < 2 and nt t 7 2. After that, we shall see which finite groups are Schur groups of these closed subsets. This way we shall establish a recognition theorem for these schemes. 

The first goal of this chapter is to show that (L) is faithfully embedded in S if L is a spherical Coxeter set having at least three elements none of them thin. The corresponding Schur groups turn out to have a Tits system. The situation will be completely described in the corresponding recognition theorem (Theorem 12.3.4). 

Let us denote by G the Schur group of L) with respect to y. Then Gyz acts transitively on zl. 

Proposition 12.3.2 Let L be a spherical Coxeter set. Assume that L has at least three elements none of them thin. Let x be an element in X, and let us denote by G the Schur group of (L) with respect to x. Then we have the following. 

---