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.     

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.     

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.     

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.     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

Some subgroups of the full linear group

Further observations are made on the author's earlier paper (Ref. Zh. Mat., 1977, 5A284) dealing with the lattice H of all subgroups of the full linear group GL(n, K) over a field K that contain the group K of diagonal matrices. It is noted, for example, that for an infinite field K all subgroups inD(n, K) are algebraic; a subgroup in H is connected if and only if it is a net subgroup; the lattice of all connected subgroups in H is isomorphic to the lattice of all marked topologies onn points; any subgroup H in H is a semidirect product H=A·H_o of a maximal connected normal subgroup H_o of H and a finite group A of, permutation matrices.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 42–46, 1977.     

Parabolic subgroups of twisted Chevalley groups over a semilocal ring

It is shown that, under minor additional assumptions, the standard parabolic subgroups of a Chevalley group G (, R) of twisted type =A_l,l odd, D_l, E₆ over a commutative semilocal ring R with involution are in one-to-one correspondence with the -invariant parabolic nets of ideals of R of type , i.e., with the sets, of ideals of R such that: (l) whenever; (2) = for all ; (3) =R for > 0. For Chevalley groups of normal types, analogous results were obtained in Ref. Zh. Mat. 1976, 10A151; 1977, 10A 301; 1978, 6A476.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 21–36, 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.     

Normal net subgroups of the full linear group

We prove that for n 3 a net subgroup of the full linear group G=GL(n, ) over an arbitrary associative ring with unity (see [1]) is normal in G if and only if it is a principal congruence subgroup. We also study the case n=2, where the situation is, in general, more complicated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 49–54, 1976.     

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.     

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.     

Parabolic subgroups of Vershik-Kerov's group

In this note we show that all parabolic subgroups of Vershik-Kerov's group (i.e. subgroups containing --the group of infinite dimensional upper triangular matrices) are net subgroups for a wide class of semilocal rings .