We give an improved bound for the order of a p-group enjoying the title property. We also point out relations between the upper central series of a p-group and of its maximal subgroups.

Szekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.

Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

Given an R-module M, the centralizer near-ring ℳR (M) is the set of all functions f: M → M with f(xr)= f(x)r for all x ∈ M and r∈R endowed with point-wise addition and composition of functions as multiplication. In general, ℳR(M) is not a ring but is a near-ring containing the endomorphism ring ER(M) of M. Necessary and/or sufficient conditions are derived for ℳR(M) to be a ring. For the case that R is a Dedekind domain, the R-modules M are characterized for which (i) ℳR(M) is a ring; and (ii)ℳR(M) = ER(M). It is shown that over Dedekind domains with finite prime spectrum properties (i) and (ii) are equivalent.

Let λ be a property that a lattice of submodules of a module may possess and which is preserved under taking sublattices and isomorphic images of such lattices and is satisfied by the lattice of subgroups of the group of integer numbers. For a ring R the lower radical Λ generated by the class λ(R) of R-modules whose lattice of submodules possesses the property λ is considered. This radical determines the unique ideal Λ (R) of R, called the λ-radical of R. We show that Λ is a Hoehnke radical of rings. Although generally Λ is not a Kurosh-Amitsur radical, it has the ADS-property and the class of Λ-radical rings is closed under extensions. We prove that Λ (Mn (R)) ⊆ Mn (Λ(R)) and give some illustrative examples.

It is shown that all the generalized Whitehead products vanish in X and all the components of XΣA have the same homotopy type when X is a T-space. It is also shown that any T-space is a G-space. The dual spaces of T-spaces are introduced and studied.

A closure operation connected with Hall subgroups is introduced for classes of finite soluble groups, and it is shown that this operation can be used to give a criterion for membership of certain special Fitting classes, including the so-called ‘central-socle’ classes.

These generalizations of Hardy's Integral Inequality are generalizations of some inequalities of B. G. Pachpatte.

We show that a commutative amenable Banach algebra need not be symmetric by constructing suitable examples.

We shall give necessary and sufficient conditions on the ring R and the group G for the group-ring RG to be a prime P. I. ring with the unique factorisation property as defined in [5].

We show that for a Henstock-Kurzweil integrable function f for every ∈ > 0 one can choose an upper semicontinuous gage function δ, used in the definition of the HK-integral if and only if |f| is bounded by a Baire 1 function. This answers a question raised by C. E. Weil.

We investigate the asymptotics and zero distribution of solutions of ω″ + Aω = 0 where A is an entire function of very slow growth. The results parallel the classical case when A is assumed to be a polynomial.

In this paper, we give a sufficient condition (Theorem) in order that one domain D1 bounded by a C2-smooth boundary can be enclosed in, or enclose, another domain D0 bounded by the same kind of boundary. A same kind of sufficient condition for convex bodies (Corollary) is also obtained.

Let X be a finite-dimensional separable metric space, presented as a disjoint union of subsets, X = A∪B. We prove the following theorem: For every prime p, c-dimZpX≦c-dimZpA + c–dimZpB + 1. This improves upon some of the earlier work by Dydak and Walsh.

The aim of this paper is to consider Problem 1 posed by Stewart and Wiegold in [6]. The main result is that if G is a finitely generated perfect group having non-trivial finite images, then there exists a finite image B of G such that the growth sequence of B is eventuallly faster than that of every finite image of G. Moreover we investigate the growth sequences of simple groups of the same order.