TY - JOUR

T1 - Subgroups of the full symplectic group containing the group of diagonal matrices. II

AU - Vavilov, N. A.

AU - Dybkova, E. V.

N1 - Copyright: Copyright 2007 Elsevier B.V., All rights reserved.

PY - 1985/7

Y1 - 1985/7

N2 - In a paper abstracted in Ref. Zh. Mat., 1981, 8A234, a description was given of the subgroups of the full symplectic group G{cyrillic}=GSp(2l,R), R being a semilocal ring, containing the group T=T(2l,R) of symplectic diagonal matrices. The study of this class of subgroups is continued in the present paper. It is proved that if R is a local ring with the residue field K and if char K≠2, and |K|≥7, then the group T is pronormal in qg. In particular, two subgroups of G{cyrillic} containing T are conjugate in G{cyrillic} if and only if they are conjugate by means of a matrix of NG{cyrillic}(T). For the field K, subgroups of GL(n,K) and Gsp(2l,K), containing a part of the group of diagonal matrices, are considered. For an almost arbitrary commutative ring, those subgroups containing T are described which are contained in the group of symplectic matrices, all elements of which below the principal diagonal belong to the Jacobson radical of the principal ring. Examples are given which show that fields containing less than 13 elements are, in fact, exceptions to the standard description of subgroups of G{cyrillic}0=Sp(2l,R), containing T ∩ G{cyrillic}0.

AB - In a paper abstracted in Ref. Zh. Mat., 1981, 8A234, a description was given of the subgroups of the full symplectic group G{cyrillic}=GSp(2l,R), R being a semilocal ring, containing the group T=T(2l,R) of symplectic diagonal matrices. The study of this class of subgroups is continued in the present paper. It is proved that if R is a local ring with the residue field K and if char K≠2, and |K|≥7, then the group T is pronormal in qg. In particular, two subgroups of G{cyrillic} containing T are conjugate in G{cyrillic} if and only if they are conjugate by means of a matrix of NG{cyrillic}(T). For the field K, subgroups of GL(n,K) and Gsp(2l,K), containing a part of the group of diagonal matrices, are considered. For an almost arbitrary commutative ring, those subgroups containing T are described which are contained in the group of symplectic matrices, all elements of which below the principal diagonal belong to the Jacobson radical of the principal ring. Examples are given which show that fields containing less than 13 elements are, in fact, exceptions to the standard description of subgroups of G{cyrillic}0=Sp(2l,R), containing T ∩ G{cyrillic}0.

UR - http://www.scopus.com/inward/record.url?scp=34250114096&partnerID=8YFLogxK

U2 - 10.1007/BF02105095

DO - 10.1007/BF02105095

M3 - Article

AN - SCOPUS:34250114096

VL - 30

SP - 1823

EP - 1832

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 1

ER -