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.     

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.     

Normalizer of the subgroup of diagonal matrices in the general linear group over a ring

The normalizer of the diagonal subgroup in the general linear group over certain classes of rings is determined. Sufficient conditions are given under which this normalizer coincides with the group of monomial matrices.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 114–118, 1983.     

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.     

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.     

Subgroups of the full linear group over a Dedekind ring

We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net of ideals of R such that, where E() is the subgroup generated by all transvections of the net subgroup G(). and is the normalizer of G(). The subgroup E() is normal in. To study the factor group we introduce an intermediate subgroup F(), E() F() G(). The group is finite and is connected with permutations in the symmetric group. The factor group G()/F() is Abelian — these are the values of a certain determinant. In the calculation of F()/E() appears the SK₁-functor. Results are stated without proof.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 13–20, 1979.     

Parabolic extensions of a Hecke ring of the general linear group

One describes a wide class of polynomials with coefficients from a Hecke ring of the general linear group, decomposing into factors over the Hecke ring of an appropriate parabolic subgroup. One gives examples of such decompositions, used in the theory of modular forms, and also the decomposition of the denominator of the Hecke series of the general linear group, yielding another proof of the rationality of the Hecke series of the group GL_n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 36–45, 1986.     

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.     

On linear groups over a field of fractions of a polycyclic group ring

LetG be a torsion free polycyclic-by-finite group andD be the field of fractions of the group algebraKG. Then any periodic subgroup ofD _n is locally finite. This answers a question posed by D. Farkas.     

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.     

Semisimilarity for matrices over a division ring

Two square matrices A and B over a ring R are semisimilar, written A$\text{?}$B, if YAX=B and XBY=A for some (possibly rectangular) matrices X, Y over R. We show that if A and B have the same dimension, and if the ring is a division ring $\text{D}$, then A$\text{?}$B if and only if A² is similar to B² and rank(A^k)=rank(B^k), k=1,2,…