Recognition theorem

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. 

In Section 6.5, we assume S to have finite valency. We shall prove that closed subsets of S which are generated by a single symmetric element of valency 2 are faithfully embedded in S. We also establish the corresponding recognition theorem. After that we shall look at closed subsets of S in which each nonidentity element has valency 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). 

In the third section of this chapter, we apply results from the two previous sections in order to prove the above-mentioned recognition theorem for spherical Coxeter sets of cardinality at least 3. 

The following theorem is the converse of Theorem 6.5.4. It says that Tits systems give rise to Coxeter sets. It makes Theorem 6.5.4 to be one of our recognition theorems. 

Morphisms are related to faithful maps, which lead naturally to the notion of a faithfully embedded closed subset. Such subsets provide an appropriate language for an attempt to establish so-called recognition theorems. These theorems deal with the question of which schemes are quotient schemes of thin schemes. We shall come back to recognition theorems and their role in scheme theory later in this preface. 

In the sixth chapter, we introduce faithful maps and faithfully embedded closed subsets. In particular, we define a closed subset to be schurian if it is faithfully embedded in itself. This chapter is also the place where we prove the first recognition theorems. 

Let us now return, as earlier promised, to a discussion on recognition theorems. 

Developing a theory of coset geometries (as mentioned in the previous footnote), one obtains from our recognition theorem of Coxeter schemes Jacques Tits main result on buildings of spherical type . This theorem asserts that each such building is associated with a group if it is thick and of rank at least 3. 

Theorem 6.4.5 and Theorem 6.5.3 are two further recognition theorems which deal with involutions. Theorem 6.4.5 relates to George Glauberman s Z -Theorem a specific class of schemes of finite valency generated by elements of valency 2. 

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