SUBGROUPS OF SLn OVER A SEMILOCAL RING

Let R be a commutative semilocal ring, G = SL(n; R) be the special linear group of degree n 3 over R and T = SD(n; R) be its subgroup of diagonal matrices. In the present paper we describe subgroups of G containing T under some mild restrictions on the residue elds of R. More precisely we prove the following result (we refer to Sections 2 o and 3 o for an explanation of all deenitions and notation used in this statement). Theorem 1. Let n 3 be a natural number and R be a semilocal ring, all of whose residue elds contain at least 3n + 2 elements. Then the standard description of subgroups of ? = SL(n; R) containing the diagonal subgroup T = SD(n; R) holds. In other words, for any such subgroup H there exists a unique D-net of ideals such that ?() H N ? (). This result generalises my earlier results, published in V3], where R was supposed to be a eld. However the proof here does not repeat the proofs presented in V3]. To the contrary, it is based on a reduction to the results of V3]. From a general viewpoint Theorem 1 is a description of overgroups of a split maximal torus in a Chevalley group of type A l. It is worth to stress the fact that unlike the case of extended Chevalley groups description of such overgroups in ordinary Chevalley groups was known only for elds. The paper is organised as follows. In 1 o we discuss previously known related results and in 2 o and 3 o we introduce the necessary notation and 1991 Mathematics Subject Classiication. 20G15, 20G35.