1.1. Introduction. In this paper, we continue with the theme of (1): the relationships holding between the Sπ (i.e. maximal π) subgroups of a locally finite group and the various local systems of that group. In (1), we were mainly concerned with ‘good’ Sπ subgroups – those which reduce into some local system (and are said to be good with respect to that system). Here, on the other hand, we are concerned with a very much more special sort of Sπ subgroup.

We recall from (3) that a group G is (centrally) eremitic if there exists a positive integer e such that, whenever an element of G has some power in a centralizer, it has its eth power. The eccentricity of an eremitic group G is the least such positive integer e.

In ((4), Theorem A) we proved that if A is a torsion free Abelian normal subgroup of a finitely generated group G with G/A nilpotent, then G has a subgroup of finite index with eccentricity 1. In this note we use a much simpler method to prove a stronger result.

In this article we study certain subclasses of the class ℒ of Lagrangian groups; that is, finite groups G having, for every divisor d of |G|, a subgroup of index d. Two such subclasses, mentioned by McLain in (6), are the class ℒ1 of groups G such that every factor group of G is in ℒ, and the class ℒ2 of groups G such that each subnormal subgroup of G is in ℒ. In section 1 we prove that a group of odd order in ℒ1 is supersoluble, and give some examples of non-supersoluble groups in ℒ1. Section 2 contains several results on the class ℒ2. In particular, it is shown that a group in ℒ2 has an ordered Sylow tower and, after constructing some examples of groups in ℒ2, a result on the rank of a group in ℒ2 is proved (Theorem 4).

0. Introduction. A group π has weak dimension (wd) ≤ n (see Cartan and Ellen-berg (2)) if Hk(π, A) = 0 for all right Z(π)-modules A and all k > n. We say that the weak dimension of a manifold M is ≤ n if wd (πl(M))≤ n. In section 1 we show that open, orientable, irreducible 3-manifolds have wd ≤ 1 if and only if they are the monotone on of 1-handle bodies. In his celebrated theorem (10), Stallings proves that finitely presented groups of cohomological dimensions ≤ 1 are free. In section 2 we prove that if π is a finitely presented group which is the fundamental group of any orientable 3-manifold with wd ≤ 1 then π is free. We also give an example to show that the finite generation of π is necessary. (Swan (11) removes the finitely presented hypothesis from Stalling's theorem.) Finally, in section 3 we generalize a theorem of McMillan (5) and prove that if M is an open, orientable, irreducible 3-manifold with finitely generated fundamental group, then M is stably (taking the product with n ≥ 1 copies of ℝ) a connected sum along the boundary of trivial (n+2)-disc Sl bundles.

Introduction. All the sets X considered in this paper will be compact convex subsets of the Euclidean plane R. We shall require the following definitions. For x, y, z ∈ R let L(x, y, x) denote the minimum of all the lengths of all connected sets containing x, y and z. Define the stretch of x with respect to X to be

and the stretch of X to be

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.

We show that if μ is a measure on the LCA group G whose Gelfand transform vanishes off the set Σ of symmetric maximal ideals, then μ µ M0(G), that is, then the Fourier-Steiltjes transform of μ vanishes at infinity. This result is then used to show μ µ L1(G)½.

We examine the possibility of establishing the existence of a compact subset E of ℝ such that E–E contains a non-degenerate interval and the semigroup generated by E has measure zero. We show that the existence of such an E is equivalent to the existence of a sequence of difference-covers that are not k-sum-covers for each k∈ℕ, and, further, is equivalent to the existence of an asymmetric maximal Raikov system.

In 1933 Borsuk(2) made the following conjecture: Every bounded set of points in Euclidean n-space En can be represented as the union of n + 1 sets of smaller diameter. He proved it for n = 2. Hadwiger (5) proved Borsuk's conjecture assuming the additional condition that the surface of the set is sufficiently smooth. On the other hand, up to now the conjecture has been proved for n ≤ 3 only (3, 4).

1. Introduction. In 1948, H. Heibronn (6) improved Vinogradov's result (11) and obtained that for any ε > 0 there exists some positive constant C(ε) which depends on ε only such that, for any real number θ and any integer N ≥ 1, there is an integer x satisfying

Here ∥t∥ means the distance from t to the nearest integer.

1. Let k be a positive integer. With the usual terminology, the series

will be said to be Riemann summable (R, k) to l if

converges for all sufficiently small x, and tends to l as x → 0. Here we take (sin nx)/(nx) as meaning 1 when n = 0. A more general summability method which has been considered by various authors ((1), (2), (5), (6), (8), (9), (10)), and is usually denoted by (ℜ, λ,k) is obtained by replacing (2) by

where λ = {λn} is a sequence of non-negative numbers increasing to ∞.

In this paper we are dealing with the oscillatory and asymptotic behaviour of the nth order, (n > 1), retarded differential equation

where g is differentiable on the half-line (t0,∞) with (I) g(t) ≤ t for every t ≥ t0 (II) g'(t) ≥ 0 for every t ≥ t0 (III) = ∞.

In a pure loss GI/G/1 queueing system, necessary and sufficient conditions are given for the output to be a renewal process. These conditions involve dependence between the service distribution and the renewal function of the arrival process: for example, if pr {service time < ξ} = 0 for some ξ > 0, then it is sufficient for the renewal function to be that of a quasi-Poisson process with index ξ.

Consider A and B to be two independent and non-degenerate random variables. It is conjectured by D. G. Kendall (private communication) that the distribution of the vector (A2, 2AB, B2) is indecomposable. The purpose of the present paper is to show that this is true.