Defining Relations for the Carpet Subgroups of Chevalley Groups over Fields

The carpet subgroups admitting a Bruhat decomposition and different from Chevalley groups are exhausted by the groups lying between the Chevalley groups of type \( B_{l} \), \( C_{l} \), \( F_{4} \), or \( G_{2} \) over various imperfect fields of exceptional characteristic 2 or 3, the larger of which is an algebraic extension of the smaller field. Moreover, as regards the types \( B_{l} \) and \( C_{l} \), these subgroups are parametrized by the pairs of additive subgroups one of which may fail to be a field and, for the type \( B_{2} \), even both additive subgroups may fail to be fields. In this paper for the carpet subgroups admitting a Bruhat decomposition we present the relations similar to those well known for Chevalley groups over fields.

     

Levi Decomposition for Carpet Subgroups of Chevalley Groups Over a Field

It is proved that a carpet subgroup of a Chevalley group of type Φ over a field is a semidirect product whose kernel is defined by a unipotent carpet of type Φ, while the noninvariant factor is a central product of carpet subgroups each of which is defined by an irreducible subcarpet of type Φi for some indecomposable root subsystem Φ _i of Φ. The obtained result can be viewed as an analog of the Levi decomposition.     

Subgroups of the Chevalley Groups and Lie Rings Definable by a Collection of Additive Subgroups of the Initial Ring

The paper provides an overview of the authors’ results on nets, carpets, elementary net subgroups, carpet subgroups, and carpet Lie rings. Some open questions are also formulated.     

Intersections of shifts of multiplicative subgroups

Using Stepanov’s method, we obtain an upper bound for the cardinality of the intersection of additive shifts of several multiplicative subgroups of a finite field. The resulting inequality is applied to a question dealing with the additive decomposability of subgroups.     

Bruhat Decomposition for Carpet Subgroups of Chevalley Groups Over Fields

Algebra and Logic - Necessary and sufficient conditions for a Bruhat decomposition to exist in a carpet subgroup of the Chevalley group over a field defined by an irreducible closed carpet of...     

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.     

On the theory of groups with generalized minimality condition for closed subgroups

We prove that a topological Abelian locally compact group with generalized minimality condition for closed subgroups is a group of one of the following types: 1) a group with minimality condition for closed subgroups, 2) an additive group of theJ _p -ring of integerp-adic numbers, 3) an additive groupR _p of the field ofp-adic numbers (p is a prime number). Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 398–409, March, 1999.     

Inverse-closed additive subgroups of fields

We describe the additive subgroups of fields which are closed with respect to taking inverses, in particular, with characteristic different from two. Any such subgroup is either a subfield or the kernel of the trace map of a quadratic subextension of the field. Partially supported by MIUR-Italy via PRIN 2003018059 “Graded Lie algebras and pro-p-groups: representations, periodicity and derivations”.     

On the quotient class of non-archimedean fields

The quotient class of a non-archimedean field is the set of cosets with respect to all of its additive convex subgroups. The algebraic operations on the quotient class are the Minkowski sum and product. We study the algebraic laws of these operations. Addition and multiplication have a common structure in terms of regular ordered semigroups. The two algebraic operations are related by an adapted distributivity law.     

Additive properties of subgroups of finite index in fields

In this paper, we confirm a conjecture of Bergelson and Shapiro concerning subgroups of finite index in multiplicative groups of fields which have maximal additive dimension. We also show that the natural extension of subgroups Gp of prime index p inside Q^? and additive dimension p+1 to the case where p is replaced by a composite integer n leads to subgroups of bounded additive dimension on a set of positive integers n of asymptotic density 1.     

Conjugacy of net subgroups of linear groups

For the full linear group over a matrix-local ring whose quotient by the Jacobson radical is not the field of two elements, we settle the question of the conjugacy ofD -net subgroups (Ref. Zh. Mat., 1977, 2A280). TwoD -net subgroups are conjugate if and only if theD -nets defining them are similar (i.e., can be transformed into each other by a permutation matrix). An analogous result is obtained forD -net subgroups of the symplectic group over a commutative local ring whose residue field contains more than three elements.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 86, pp. 11–18, 1979.     

Subgroups normalized by the elementary Levi subgroup

Subgroups of the unipotent radical of a maximal parabolic subgroup of a Chevalley group over a field K, which are normalized by the commutator subgroup of the Levi subgroup, are described. It is shown that in the typical case, such subgroups are in one-to-one correspondence with the closed subsets of {1, 2, ..., n} for a natural n. In the exceptional cases the classification also involves additive subgroups of K. See the table in the paper for a detailed list of possibilities. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 199–215.     

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.     

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.     

一类含参变量的Sierpinski垫片的Hausdorff测度

Sierpinski垫片是具有严格自相似性的经典分形集之一.本文给出了一类含参变量的Sierpinski垫片.通过它在x轴上的投影估计了这类Sierpinski垫片的Hausdorff测度的下界,然后精心构造了一个仿射变换,将参变量的范围由(0,π/3)的讨论转换到(π/3,π)的讨论,从而得到了这类Sierpinski垫片的Hausdorff测度的精确值.     

Closed Elementary Nets over a Field of Characteristic 0

Siberian Mathematical Journal - Under study are the questions of closedness of an elementary net (carpet) over a field. Every complete net is closed (admissible). We construct an example of...     

Nil properties for rings which are sums of their additive subgroups

Suppose that a ring R is a direct sum of a finite number of its additive subgroups, and the union of these subgroups is closed under multiplication. We show that if all rings among these subgroups are nilpotent (left T-nilpotent, locally nilpotent or Baer radical), then the whole ring R satisfies the same property.     

Homotopically non-trivial additive subgroups of Hilbert spaces

We prove that for a line-free closed additive subgroup of a Hilbert space certain orthogonal projections lead to coverings of this group. This makes it possible to obtain additive subgroups which are homotopically non-trivial.