On decomposability of finite groups

Let G be a finite group. A normal subgroup N of G is a union of several G-conjugacy classes, and it is called n-decomposable in G if it is a union of n distinct G-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.     

Some criteria for supersolubility in products of finite groups

Let H and T be subgroups of a finite group G. H is said to be permutable with T in G if HT = TH. In this paper, we use the concept of permutable subgroups to give two new criterions of supersolubility of the product G = AB of finite supersoluble groups A and B.      

Fourier series criteria for operator decomposability

Let U be an invertible operator on a Banach space Y. U is said to betrigonometrically well-bounded provided the sequence {Uⁿ} _n =– is the Fourier-Stieltjes transform of a suitable projection-valued function E(·): [0, 2](Y). This class of operators is known to apply naturally to a variety of classical phenomena which exclude the presence of spectral measures. In the case Y reflexive we use the Cesáro means _n(U, t) of the trigonometric series _k0 k^–e^iktU^k, whichformally transfers the discrete Hilbert transform to Y, in order to give three separate necessary and sufficient conditions for U to be trigonometrically well-bounded. One of these conditions is sup {_n(U,t): n 1, t [0,2]} <      

On decidability of the decomposability problem for finite theories

We consider the decomposability problem for elementary theories, i.e. the problem of deciding whether a theory has a nontrivial representation as a union of two (or several) theories in disjoint signatures. For finite universal Horn theories, we prove that the decomposability problem is $ \sum _1^0 $ \sum _1^0 -complete and, thus, undecidable. We also demonstrate that the decomposability problem is decidable for finite theories in signatures consisting only of monadic predicates and constants.     

Some presentations for the special linear groups on finite fields

We simplify the Steinberg presentation of SL _n (F _d ), where n≥1 and F _d is any finite field with d elements. That presentation has the elementary matrices e _ij (r), with i,j∈{1,...,n}, i≠=j and r∈F _d , as generators, and (E1)–(E3), described at the opening of this work, as relations. The presentation that we shall obtain reduces the number of generators e _ij (r) and relations (E1)–(E3). In particular, relations (E3) are considerably reduced. Received: March 16, 1998; in final form: November 3, 2000?Published online: October 2, 2001     

Some inequalities for the Baer-invariant of a pair of finite groups

In this paper we introduce the concept of Baer-invariant of a pair of groups with respect to a variety of groups v. Some inequalities for the Baer-invariant of a pair of finite groups are obtained, when v is considered to be the Schur-Baer variety. We also present a condition for which the order of the Baerinvariant of a pair of finite groups divides the order of the Baer-invariant of their factor groups. Finally, some inequalities for the Schur-multiplier of a pair of finite nilpotent groups and their factor groups are given.     

Some results on supersolvability of finite groups

We get new characterizations of finite supersolvable groups based on the structure of the generalized Fitting subgroup. Then we extend our results to the saturated formations containing the class of all supersolvable groups. Authors’ addresses: M. Asaad, Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt; Piroska Cs?rg?, Department of Algebra and Number Theory, E?tv?s University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary     

Some results on the generalized hypercenter of finite groups

We study the structure of a finite group G under the assumption that certain subgroups lie in the generalized hypercenter of G.     

Some finite groups with large conjugacy classes

We derive some properties of a family of finite groups, which was investigated by Camina, Macdonald, and others. For instance, we give information about the Schur multipliers of the class twop-groups in this family. A large part of this paper was written while the author was visiting the Department of Mathematics of the University of Trento. The author is indebted to this department, and in particular to C.M. Scoppola, for their kind hospitality. The author is also grateful to D. Chillag for his constructively destructive criticism of the first version of this paper.     

Some soluble groups of finite rank and some related matrix groups

Let G be a torsion-free soluble group of finite rank and F any field. The group algebra FG is an Ore domain; let D denote its division ring of quotients. It seems likely that D is always locally residually finite-dimensional over F. This is certainly so in the non-modular case. Here in some special situations we settle the modular case. We include some applications to groups of matrices.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 894–901, July–August, 1991.     

Some aspects of groups acting on finite posets

Let P be a finite poset and G a group of automorphisms of P. The action of G on P can be used to define various linear representations of G, and we investigate how these representations are related to one another and to the structure of P. Several examples are analyzed in detail, viz., the symmetric group $\text{G}$_n acting on a boolean algebra, GL_n(q) acting on subspaces of an n-dimensional vector space over GF(q), the hyperoctahedral group B_n acting on the lattice of faces of a cross-polytope, and $\text{G}$_n acting on the lattice Π_n of partitions of an n-set. Several results of a general nature are also proved. These include a duality theorem related to Alexander duality, a special property of geometric lattices, the behavior of barycentric subdivision, and a method for showing that certain sequences are unimodal. In particular, we give what seems to be the simplest proof to date that the q-binomial coefficient $\text{k+l}\text{k}$ has unimodal coefficients.     

Multiple decomposability of probabilities on contractible locally compact groups

Operator decomposable probabilities on vector spaces – generalizing (semi-)stable and self-decomposable laws – are well known. Here we are concerned with multiple decomposability on locally compact groups. In fact, as it turned out that contraction properties play an essential role, throughout we concentrate on contractible locally compact groups.