The purpose of this expository article is to familiarize the reader with one of the fundamental problems in the theory of infinite groups. We give an up-to-date account of the so-called Fox problem which concerns the identification of certain normal subgroups of free groups arising out of certain ideals in the free group rings. We assume that the reader is familiar with the elementary concepts of algebra.

Let G be a lattice-ordered group (l-group), and let t, u∈ G+ . We write tπu if t ∧ g = 1 is equivalent to u ∧ g = 1, and say that a tight Riesz order T on G is π-full if t ∈ T and t π U imply u∈T. We study the set of π-full tight Riesz orders on an l-permutation group (G, Ω), Ω a totally ordered set.

Suppose throughout that {sn} is a sequence of real numbers, λ > - 1, a > 0, and β is real. Let N be any non-negative integer such that αN + β > l.

We are concerned primarily with the logarithmic summability method L and the Borel-type method (B, α, β). Some known results involve the Abel-type summability method Aγ.

Banchoff and Pohl [3] have proved the following generalization of the isoperimetric inequality.

Theorem. If γ is a closed, not necessarily simple, planar curve of length L, and w(p) is the winding number of a variable point p with respect to γ, then

1

with equality holding if and only if γ is a circle traversed a finite number of times in the same sense.

In this paper define the spaces l ∞(Δ), c(Δ), and c0(Δ), where for instance l∞(Δ) = {x=(xk):supk |xk -xk + l|< ∞}, and compute their duals (continuous dual, α-dual, β-dual and γ-dual). We also determine necessary and sufficient conditions for a matrix A to map l ∞(Δ) or c(Δ) into l∞ or c, and investigate related questions.

A compactification αX of the space X is called an n -point compactification if the remainder αX — X consists of exactly n points. K. D. Magill [5] showed that if Y has an n-point compactification and if f:X→ f(x) = Y is a compact continuous mapping of the space X onto Y, then X also has an n-point compactification.

The invariant operator range lattices of a wide class of uniformly closed algebras (including C *-algebras) are stable under weak closures. There is an algebra whose invariant operator range lattice contains properly the corresponding lattice of its norm closure. An operator range transitive algebra is operator range n -transitive for all n. A normal operator is algebraic if and only if each of its invariant operator ranges is the range of some operator commuting with it.

In [8] and subsequent papers, Jônsson (et al) developed a lattice identity which reflects precisely Desargues Law in projective geometry in that a projective geometry satisfies Desargues Law if and only if the geometry, qua lattice, satisfies this identity. This identity, appropriately called the Arguesian law, has become exceedingly important in recent investigations in the variety of modular lattices (see for example [2], [3], [9], and [12]). In this note, we supply two possible lattice identities for the Pappus' Law of projective geometry.

In [9] de la Torre proved that if is a finite measure space and T is a linear operator on a real for some fixed p, 1 < p < ∞ , such that ||T||P ≤ 1 and simultaneously ||T||∞ ≤ l, and also such that there exists with Th = h and h≠0 a.e., then the dominated ergodic theorem holds for T, i.e. for every we have

de la Torre proved his result, by showing that the operator S, defined by Sf = (sgn h) - T(f • sgn h) for is positive, and by applying Akcoglu's theorem [1] to S.

The inversion and the characterization of the convolution transform is derived via the concept of unimodality introduced by Khintchine (1938). This method yields simple and intuitively appealing proofs.

Un groupe abélien est dit super-décomposable si il ne possède pas de facteur direct indécomposable non nul. Les groupes super-décomposables doivent être nécessairement sans-torsion et un premier exemple en fut donné par A. L. S. Corner [1]. Un exemple différent apparait dans P. A. Krylov [4]. De plus P. Jambor et J. Bečvar, ont développé, dans [3] une généralisation de la notion de sous-groupe de base dans laquelle apparait de façon naturelle des groupes super-décomposables. Dans cette note, nous donnons d'abord un critère pour qu'un groupe soit super-décomposable et nous construisons ensuite une famille de groupes qui satisfont à ce critère, donnant ainsi de nouveaux exemples de groupe super-décomposable.

A set of sentences T is called independent if for every . It is countably independent if every countable subset is independent. In flnitary first order logic, L ωω, the two notions coincide because of compactness. This is not the case for infinitary logic.

Let β(n) = ∑p|nP and B(n) = ∑Pα ||nαP denote the sum of distinct prime divisors of n and the sum of all prime divisors of n respectively. Both β(n) and B(n) are additive functions which are in a certain sense large (the average order of B(n) is π2n/(6 log n), [1]).

Suppose R is a commutative ring with identity. Let M be an R -module, and suppose f is a function from M to R. How do we characterize the property that f be a quadratic form?

In the category of T1 -spaces and continuous maps, every space X has a unique maximal essential extension. X has an injective hull iff card X≤1.

By an ortholattice we mean a lattice with 0 and 1 and a complementation operation which is an involutorial antiautomorphism. The free modular ortholattice on two generators has 96 elements—cf. J. Kotas [8].

Our aim is to compute for all n > 2, ψ(n, h), the number of components of a certain quotient of the fixed point set of an involution in the "mod-n" Teichmuller space. This answers part of a question raised by Earle [2] and corrects and extends the answer due to Zarrow (See Theorem 2 of [6]).

Mason, [1, Theorem 2.6], proved that for any near-ring R, there are no non-trivial injective R-modules. In his proof he embedded R into a simple R -module G of arbitrarily high cardinality.

In this note we prove:

Theorem. Let R be a right primitive ring with pair-wise non-isomorphic faithful irreducible modules Ml, M2,…, Mk. Let Di = EndR Mi. For each i, let be elements of Mi linearly independent over Di. For each i, let be a set of elements of Mi. Then there exists an element r of R such that uij = vijr, for i = 1, 2, …, k and j = 1, 2, …, ni.