The introduction of curvature considerations in the past decade into Combinatorial Group Theory has had a profound effect on the study of infinite discrete groups. In particular, the theory of negatively curved groups has enjoyed significant and extensive development since Cannon's seminal study of cocompact hyperbolic groups in the early eighties [7]. Unarguably the greatest influence on the direction of this development has been Gromov's tour de force, his foundational essay in [12] entitled Hyperbolic Groups. Therein Gromov hints at the prospect of developing a corresponding theory of “non-positively curved groups” in his non-definition (Gromov's terminology) of a semihyperbolic group as a group that “looks as if it admits a discrete co-compact isometric action on a space of nonpositive curvature”; [12, p. 81]. Such a development is now occurring and is closely related to the other notable outgrowth of the theory of negatively curved groups, that of automatic groups [10]; we mention here the works [3] and [6] as developments of a theory of nonpositively curved groups along with Chapter 6 of Gromov's more recent treatise [13]. A natural question that serves both to guide and organize the developing theory is: to what extent is the well-developed theory of negatively curved groups reflected in and subsumed under the developing theory of nonpositively curved groups? Our overall interest is in one aspect of this question—namely, as the question relates to the boundaries of groups and spaces: can one define the boundary of a nonpositively curved group intrinsically in a way that generalizes that of negatively curved groups and retains some of their essential features?

McCoy proved that for a right ideal A of S = R[x1, . . ., xk] over a ring R, if rS(A) ≠ 0 then rR(A) ≠ 0. We extend the result to the Ore extensions, the skew monoid rings and the skew power series rings over non-commutative rings and so on.

We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic subgroup of GL2 (defined by F. Momose). We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.

We consider associative algebras $\Lambda$ over a field provided with a direct sum decomposition of a two-sided ideal $M$ and a sub-algebra $A$ – examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of $\Lambda$. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.

Let G be a polycyclic-by-finite group and let K[G] denote its group algebra over the field K. In this paper we discuss localization in K[G] and in particular we prove that every faithful completely prime ideal is localizable. Furthermore, using a sequence of localizations, we show that, for G polyinfinite cyclic, the classical right quotient ring (K[G]) is in fact a universal field of fractions for K[G]. Finally we offer an example of a domain K[G] which does not have a universal field of fractions.

Let G be a finite group, (ZG) the group of units of the integral group ring ZG and 1(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively (ZG) for particular groups G. This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described (ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of order 16 were discussed in their paper. Our main aim is to complete the description of (ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case.

In what follows, groups are written additively, and commutators are denoted by brackets:

A group is metabelian if it satisfies the law

It is conceivable, though not plausible, that this law is equivalent to a law, or a set of laws, in only three variables, or even two. The present note shows that this is not the case.

We establish the mapping properties of Fourier-type transforms on rearrangement-invariant quasi-Banach function spaces. In particular, we have the mapping properties of the Laplace transform, the Hankel transforms, the Kontorovich-Lebedev transform and some oscillatory integral operators. We achieve these mapping properties by using an interpolation functor that can explicitly generate a given rearrangement-invariant quasi-Banach function space via Lebesgue spaces.

We give a complete description of the Brown–McCoy radical of a semigroup ring R[S], where R is an arbitrary associative ring and S is a commutative cancellative semigroup; in particular we obtain the answer to a question of E. Puczyłowski stated in [11]

Throughout this note all rings R are associative with unity 1; all semigroups S are commutative and cancellative with unity. Note that the condition that R and S have a unity can be dropped (cf. [8]). The quotient group of S is denoted by Q(S). We say that S is torsion free (resp. has torsion free rank n) if Q(S) is torsion free (resp. has torsion free rank n). The Brown–McCoy radical (i.e. the upper radical determined by the class of all simple rings with unity) of a ring R is denoted by u(R). We refer to [2] for further detail on radicals and in particular on the Brown–McCoy radical.

First we state some well-known results and a preliminary lemma. Let R and T be rings with the same unity such that R ⊂ T. Then T is said to be a normalizing extension of R if T = Rx1+…+Rxn for certain elements x1, …, xn of T and Rxi = xiR for all i such that 1 ∨i∨n. If all xi are central in T, then we say that T is a central normalizing extension of R.

Let G be a group and C = [G, G] be its commutator subgroup. Denote by c(G) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. The exact values of c{G) are computed when G is a free nilpotent group or a free abelian-by-nilpotent group. If G is a free nilpotent group of rank n>2 and class c>2, c(G) = [n/2] if c = 2 and c(G) = n if c>2. If G is a free abelian-by-nilpotent group of rank n > 2 then c(G) = n.

Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces.

Let G be a graph with p points and q lines, and genus γ. The thicknesst(G) has been introduced as the minimum number of planar subgraphs whose union is G. This topological invariant of a graph has been studied by Battle, Harary and Kodama [1], Tutte [7], Beineke, Harary and Moon [3], and Beineke and Harary [2].

It is natural to generalise this concept of the thickness of a graph to the union of graphs with a specified genus. We say that the n-thickness of G is the minimum number of subgraphs of genus at most n whose union is G. Denoting the n-thickness of G by tn, we write in particular t0 the 0-thickness, i.e., the thickness.

We study moduli spaces of framed perverse instantons on ℙ3. As an open subset, it contains the (set-theoretical) moduli space of framed instantons studied by I. Frenkel and M. Jardim in [9]. We also construct a few counter-examples to earlier conjectures and results concerning these moduli spaces.

We let X be an arbitrary infinite set. A semigroup S of total or partial transformations of X is called -normal if hSh-1 = S, for all h in , the symmetric group on X. For example, the full transformation semigroup , the semigroup of all partial transformations , the semigroup of all 1–1 partial transformations and all ideals of and are -normal.

Let G be a group given in terms of generators and denning relations. The order problem is said to be solvable for (the given presentation of) the group G if, given any element W of G (as a word in the given generators of G), we can determine the order of W in G. The power problem is solvable for G if, given any pair X, Y of elements of G, we can determine whether or not X belongs to the cyclic subgroup {Y} of G generated by Y. It is easy to see that if either of these problems is solvable for G, then the word problem is also solvable for G.

The Dedekind η-function is defined by

where τ lies in the upper half plane ℋ = {tau;|Im(τ) > 0}, and x = e2πiτ. It is a modular form of weight ½ with a multiplier system. We define an η-product to be a function f (τ) of the form

where rδ ε ℤ. This is a modular form of weight with a multiplier system. The Fourier coefficients of η-products are related to many well-known number-theoretic functions, including partition functions and quadratic form representation numbers. They also arise from representations of the “monster” group [3] and the Mathieu group M24 [13]. The multiplicative structure of these Fourier coefficients has been extensively studied. Recent papers include [1], [4], [5] and [6]. Here we study the connections between the density of the non-zero Fourier coefficients of f(τ) and the representability of f(τ) as a linear combination of Hecke character forms (defined in Section 4 below). We first make the following definition.

Throughout the paper we consider only finite groups.

J. C. Beidleman and H. Smith [3] have proposed the following question: “If G is a group and Ha subnormal subgroup of G containing Φ(G), the Frattini subgroup of G, such that H/Φ(G)is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.

Our aim in this paper is to deal with Sobolev's embeddings for Sobolev–Orlicz functions with ∇u ∈ Lp(·) logLq(·)(Ω) for Ω ⊂ n. Here p and q are variable exponents satisfying natural continuity conditions. Also the case when p attains the value 1 in some parts of the domain is included in the results.

Throughout the paper, rings are associative rings with identity. A ring is called right duo if every right ideal is two-sided, and it is called right p.p. if every principal right ideal is projective. A left duo (p.p.) ring is denned similarly, and a duo (p.p.) ring will mean a ring which is both right and left duo (p.p.). There is a right p.p. ring that is not left p.p. (see Chase [2[). Small [9] proved that right p.p. implies left p.p. if there are no infinite sets of orthogonal idempotents, and Endo [5, Proposition 2] has shown the same implication in the case where each idempotent in the ring is central. Since Courter [3, Theorem 1.3] noted that every idempotent in a right duo ring is central, we can simply speak of right duo p.p. rings. A typical example of a right duo ring which is not left duo is the following. Let F be a field and F(x) the field of rational functions over F. Let R = F(x)× F(x) as an additive group and define the multiplication as follows:

Then R is a local artinian ring with c(RR) = 2 and c(RR)= 3. Thus R is right duo but not left due.

Motivated by a relation on submodules of a module used by both A. W. Goldie and P. F. Smith, we say submodules X, Y of M are β* equivalent, Xβ*Y, if and only if is small in and is small in . We show that the β* relation is an equivalence relation and has good behaviour with respect to addition of submodules, homomorphisms and supplements. We apply these results to introduce the class of -supplemented modules and to investigate this class and the class of H-supplemented modules. These classes are located among various well-known classes of modules related to the class of lifting modules. Moreover these classes are used to explore an open question of S. H. Mohamed and B. J. Mueller. Examples are provided to illustrate and delimit the theory.