Subgroups,of Chevalley groups over a locally finite field,defined by a family of additive subgroups

It is proved that every elementary carpet of nonzero additive subgroups which is associated with a Chevalley group of a Lie rank exceeding one over a locally finite field coincides, up to conjugation by a diagonal element, with a carpetwhose additive subgroups are equal to some chosen subfield of the ground field. A similar result is obtained for a full matrix carpet (a full net).     

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.     

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.

     

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.     

DECIMAL EXTENSIONS OF THE ADDITIVE GROUP OF INTEGERS

We introduce the concept of decimal extension of the additive group of integers. We characterize these extensions as the linearly ordered abelian groups containing at least one Archimedean element. Finally, we study the additive group of rationals and its subgroups as particular cases of decimal extension.     

Closed nets in linear groups

The paper reviews the author’s results in the theory of elementary nets (carpets). In particular, closed (admissible) nets are investigated. For an elementary net (a net considered without the diagonal) of additive subgroups of an arbitrary commutative ring, the concepts of the derivative net, the closure of the net, and the net associated with the elementary group are introduced. Factorization of the elementary groups is proposed. This factorization is then used to construct an example of a closed (admissible) net which cannot be completed to a (complete) net. For a third-order elementary net σ of additive subgroups of a commutative ring, decomposition of an elementary transvection from the elementary group E(σ) is obtained.     

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.     

Medial and distributively generated near-rings

In this paper we note some properties of fully invariant additive subgroups of near-rings and apply these results to d.g., medial, or subdirectly irreducible near-rings     

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.

     

The structure of lattice-ordered commutative topological groups of compact origin

It is shown that every lattice-ordered commutative separable topological group of compact origin can be obtained from a finite number of its linearly ordered subgroups, each of which is isomorphic either to the additive group of real numbers with the natural topology and the usual order or to a subgroup of the additive group of real numbers with the discrete topology and the usual order, admitting a finite system of linearly independent generators, by forming in turn the direct and the lexicographic products.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 855–860, June, 1977.     

Generalised linear model trees with global additive effects

Model-based trees are used to find subgroups in data which differ with respect to model parameters. In some applications it is natural to keep some parameters fixed globally for all observations while asking if and how other parameters vary across subgroups. Existing implementations of model-based trees can only deal with the scenario where all parameters depend on the subgroups. We propose partially additive linear model trees (PALM trees) as an extension of (generalised) linear model trees (LM and GLM trees, respectively), in which the model parameters are specified a priori to be estimated either globally from all observations or locally from the observations within the subgroups determined by the tree. Simulations show that the method has high power for detecting subgroups in the presence of global effects and reliably recovers the true parameters. Furthermore, treatment–subgroup differences are detected in an empirical application of the method to data from a mathematics exam: the PALM tree is able to detect a small subgroup of students that had a disadvantage in an exam with two versions while adjusting for overall ability effects.

     

Integral mean values of modular L-functions

We consider the mean squares of L-functions associated to modular forms with respect to Hecke congruence subgroups, expressing the mean value as an inner product. This avoids the discussion of generalized additive divisor problems. As applications, we obtain asymptotic formulas for both weighted and unweighted mean squares.     

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.     

Near integral domains II

A direct product decomposition is given for the multiplicative semigroup of a finite near integral domain in terms of the subsemigroup of left identities and a group of automorphisms on the additive group of the domain. Conditions are given which insure that every element will have a uniquen-th root. If there existsx≠0 such that (?x)y=?(xy), for eachy, then the additive group of the near integral domain is abelian. Other conditions sufficient for the commutativity of the additive group are given. An example illustrates that non-isomorphic finite near integral domains can have a left ideal decomposition into Sylow subgroups which are isomorphic as near-rings. Another example shows that an infinite near integral domain need not have a nilpotent additive group, even in the d. g. case. It is conjectured that for each natural numbern there is a near integral domain whose additive group is of nilpotent classn.     

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”.     

矩阵环的有限乘法子群

将Minkowski关于有限整数矩阵群的著名结果推广到一般的环上,主要结果是证明了:对任意环R,如果R的加法群为有限生成的自由Abel群,则R的所有乘法可逆元构成的群U(R)中的有限子群精确到同构只有有限多个.     

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.     

Affine Module Groups and Their Automorphisms

An affine module group is a semidirect extension of an additive module group by its automorphism group. Maximal Abelian normal subgroups of an affine group are described. It is proved that operator isomorphisms of affine groups are induced by module automorphisms. Automorphisms of an affine group which do not leave a module fixed are treated. And conditions are specified for a module to be non-characteristic in its affine group.     

On the average number of subgroups of the group $${\mathbb {Z}}_m \times {\mathbb {Z}}_n$$ Z m × Z n with k-th power sequence

The Ramanujan Journal - Let $$\mathbb {Z}_{n}$$ be the additive group of residue classes modulo n. Let s(m, n) denote the total number of subgroups of the group $$\mathbb {Z}_{m} \times...     

On graded Thierrin radicals of graded rings

In this paper, we study the graded Thierrin radical and the classical Thierrin radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous elements is again homogeneous. There are two versions of this graded radical, the graded Thierrin and the large graded Thierrin radical. We establish several characterizations of the graded Thierrin radical and prove that the largest homogeneous ideal contained in the classical Thierrin radical of a graded ring coincides with the large graded Thierrin radical of that ring.