TY - JOUR

T1 - Subgroups of the split orthogonal group. II

AU - Vavilov, N. A.

N1 - Copyright: Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2002

Y1 - 2002

N2 - In the first paper of the present series, we proved the standardness of subgroups containing a split maximal torus in the split orthogonal group SO(n, R) over a commutative semilocal ring R for the following two situations: 1) n is even; 2) n is odd and R = K is a field. In the present paper, we prove the standardness of intermediate subgroups over a semilocal ring R in the case of an odd n. Together with the preceding papers by Z. I. Borewicz, the author, and E. V. Dybkova, this paper completes the description of the overgroups of split maximal tori in the classical groups over semilocal rings. The analysis of odd orthogonal groups turned out to be technically much more difficult than other classical cases. Unlike all preceding papers, the proof of the key step in the reduction to the case of a field relies on calculations in a class of semisimple elements that are neither microweight elements nor semisimple root elements.

AB - In the first paper of the present series, we proved the standardness of subgroups containing a split maximal torus in the split orthogonal group SO(n, R) over a commutative semilocal ring R for the following two situations: 1) n is even; 2) n is odd and R = K is a field. In the present paper, we prove the standardness of intermediate subgroups over a semilocal ring R in the case of an odd n. Together with the preceding papers by Z. I. Borewicz, the author, and E. V. Dybkova, this paper completes the description of the overgroups of split maximal tori in the classical groups over semilocal rings. The analysis of odd orthogonal groups turned out to be technically much more difficult than other classical cases. Unlike all preceding papers, the proof of the key step in the reduction to the case of a field relies on calculations in a class of semisimple elements that are neither microweight elements nor semisimple root elements.

UR - http://www.scopus.com/inward/record.url?scp=52649181188&partnerID=8YFLogxK

U2 - 10.1023/A:1020378516076

DO - 10.1023/A:1020378516076

M3 - Article

AN - SCOPUS:52649181188

VL - 112

SP - 4266

EP - 4276

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 3

ER -