Subgroups of the full linear group over a semilocal ring

Let be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal idealmC contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals such that G()HN(), whereN() is the normalizer of the D-net subgroup . A transparent classification of subgroups GL(n, ) normalizable by diagonal matrices is thus obtained. Further, the factor groupN()/G() is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 32–34, 1978.     

Parabolic subgroups of the full linear group over a Dedekind ring of arithmetical type

Suppose K is a global field, S a finite set of valuations of K containing all Archimedean valuations, and R the ring of S-integral elements of K. Assume that card S2, R is generated by its invertible elements, and the ideal of R generated by the differences –1 for all invertible coincides with R. Under these assumptions, the parabolic subgroups of GL(n, R) are described. Namely, for each parabolic subgroup P there exists a unique net of ideals of R (Ref. Zh. Mat., 1977, 2A280) such that e()PG(), where G is the net subgroup of () and E() is the subgroup generated by the transvections in G(). It is shown that E() is a normal subgroup of G(). The factor group G(/E()) is studied. The case of the special linear group is also considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 66–79, 1977.In conclusion,the author would like to thank Z. I. Borevich for posing the problem and for his continued interest, and A. A. Suslin for useful discussions leading to simplifications of some of the proofs.     

Subgroups of the full linear group over the field of quotients of a principal ideal domain

The structure is described of all subgroups of the full linear group over the field of quotients of a principal ideal domain R which contain the special linear group over R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 86, pp. 185–187, 1979.     

The index of a net subgroup of a symplectic group over a Dedekind ring

An explicit formula is derived for the index of a net congruence subgroup of a symplectic group over a Dedekind ring. A classification of symplectic D-nets over a field is obtained along the way.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 74–86, 1978.     

Subgroups between special linear groups over a ring and its subring

All the subgroups between the special linear groups SL(n, o) and SL(n, o) are described in the following two cases: 1) o is a real-closed field, and o is its algebraic closure; 2) o is a Euclidean ring, and o is its quotient field.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 335–345, September, 1969.     

Self-normalizing nilpotent subgroups of the full linear group over a finite field

It has been proved (Ref. Zh. Mat., 1977, 4A170) that in the full linear group GL(n,q), n=2, 3, over a finite field of q elements, q odd or q=2, the only self-normalizing nilpotent subgroups are the normalizers of Sylow 2-subgroups and that for even q>2 there are no such subgroups. In the present note it is deduced from results of D. A. Suprunenko and R. F. Apatenok (Re. Zh. Mat., 1960, 13586; 1962, 9A150) that this is true for any n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 86, pp. 34–39, 1979.     

On the normal structure of the general linear group over a ring

The present paper is devoted to the study of normal subgroups of the general linear group over a ring and the centrality of the extension St (n, R) E(n,R). The notions of the standard commutator formula and the standard normal structure of GL(n, R), E(n, R), and St (n,R) and their relationships are discussed. In particular, it is shown that the normality of E(n, R) in GL (n,R) and the standard distribution of subgroups normalized by E (n, R) follow from some conditions of linear dependence in R. Also, it is proved that the standardness of the normal structure of GL (n,R) and the centrality of K₂(n, R) in St (n,R) follow from the same conditions over a quotient ring R/I, provided that si In}-1. Under certain additional assumptions (for example, I is contained in the Jacobson radical of R), the converse is also true. The standard technique due to H. Bass, Z. I. Borevich, N. A. Vavilov, L. N. Vaserstein, W. van der Kallen, A. A. Suslin, M. S. Tulenbaev, and others is used and developed in this paper. Bibliography: 21 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 166–182.The author gratefully acknowledges the support of the St. Petersburg Mayor's Office for the grant for young scientists and of the SFB 343 in Bielefeld University. The investigation presented in this paper was made possible in part by grant No. JHP100 from the International Science Foundation and the Russian Government.     

Subnormalizer of net subgroups in the general linear group over a ring

Let be a commutative ring in which the elements of the form ²–1, * generate the unit ideal and assume that a is any D-net of ideals of of order n. It is shown that the normalizerN() of the net subgroup G() (RZhMat, 1977, 2A280) coincides with its subnormalizer in GL(n, ). For noncommutative the corresponding result is obtained under the assumptions: 1) in the elements of the form — 1, where runs through all invertible elements of the center of , generate the unit ideal, and 2) the subgroup G() contains the group of block diagonal matrices with blocks of order 2.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 14–19, 1982.     

Products of reflections in the general linear group over a division ring

Let F be a division ring and A?GL_n(F). We determine the smallest integer k such that A admits a factorization A=R₁R₂?R_k?1B, where R₁,…,R_k?1 are reflections and B is such that rank(B?I_n)=1. We find that, apart from two very special exceptional cases, k=rank(A?I_n). In the exceptional cases k is one larger than this rank. The first exceptional case is the matrices A of the form I_m⊕αI_n?m where n?m?2, α≠?1, and α belongs to the center of F. The second exceptional case is the matrices A satisfying (A?I_n)²=0, rank(A?I_n)?2 in the case when char F≠2 only. This result is used to determine, in the case when F is commutative, the length of a matrix A?GL_n(F) with detA=±1 with respect to the set of all reflections in GL_n(F).