Induced characters of the projective general linear group over a finite field

Using a general result of Lusztig, we find the decomposition into irreducibles of certain induced characters of the projective general linear group over a finite field of odd characteristic.     

Infinite chains of successive normalizers in a general linear group over a field

Let K be a field of zero characteristic or a field of characteristic two and transcendence degree at least one. It is shown that in a general linear group GL(n,K) of arbitrary degree n there exist subgroups for which the ascending chains of successive normalizers are infinite. All the factors of the constructed chains are second order groups.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk, Vol. 191, pp. 44–48, 1991.     

Maximal solvable subgroups of the special linear group over an arbitrary field

Minsk. Translated fromSibirski Matematicheski Zhurnal, Vol. 33, No. 6, pp. 39–46, November–December, 1992.     

On parabolic induction on inner forms of the general linear group over a non-archimedean local field

We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form \(\pi \otimes \sigma \) where \(\pi \) is a ladder representation and \(\sigma \) is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual.     

Isomorphisms of general linear groups over associative rings graded by an Abelian group

In this paper, we give a simpler proof of the Golubchik–Mikhalev–Zelmanov theorem on the structure of isomorphisms between general linear groups over associative rings, and also prove an extension of this theorem for linear groups over rings graded by an Abelian group.     

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.     

On the invertible algebras linear over an Abelian group

In this paper using the second order formula, namely the ??(?)-identities, we characterize some subclasses of the invertible algebras that are linear over an Abelian group and have restrictions on the use of the automorphisms of the corresponding group.     

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).     

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.     

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 Sylow subgroups of the general linear group over a complete discrete valuation ring

We derive necessary and sufficient conditions for conjugacy of Sylow subgroups in the full linear group over the ring of all integers of a finite extension of the field of p-adic numbers Q_p, p 2.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 918–924, July–August, 1991.