The problem of locating and classifying the normal subgroups of GLn(R), the general linear group of degree n over a commutative ring R with an identity element, has received considerable attention. The solution when R is a field is well known (of. Dieudonné(5), Artin(1)): unless n is equal to two and R has two or three elements, normal subgroups of GLn(R) either lie in the centre of GLn(R) or contain the special linear group SLn(R). However, if R is not a field, then for each ideal I of R the natural map R → R/I induces a homomorphism

and, if 0 < I < R, the kernel of θ1 is a non-central normal subgroup of GLn(R) which does not contain SLn(R). The most that may be expected is that each normal subgroup determines an ideal I of R, in such a way that all normal subgroups determining the same ideal I lie between suitably defined greatest and smallest normal subgroups of GLn(R) corresponding to I. For example, write ZI for the inverse image of the centre of GLn(R/I) under the homomorphism θI, and write KI for the intersection of SLn(R) with the kernel of θI. Then KI ≽ ZI, and the results of Klingenberg(7) and Mennicke(9) show that if n ≥ 3, and if R is either a local ring or the ring of rational integers, then any normal subgroup H of GLn(R) satisfies

for some uniquely determined ideal I. There are many similar theorems. That the above result breaks down for arbitrary rings, even for large n, follows easily from the negative solution to the congruence subgroup problem for certain rings of algebraic integers (see Bass, Milnor and Serre(3)).

In this paper upper bounds on the Engel length of a group G of prime-power exponent are obtained, under varying conditions on a normal subgroup H and the quotient G/H. Examples are constructed to show that the bounds are best possible when the nilpotency class of H is less than the relevant prime.

We shall be working throughout this paper in the class of locally finite groups introduced in (3) and further discussed in (5) and (6), and all groups appearing will be assumed to belong to this class. By definition, is the largest subgroupclosed class of locally finite groups satisfying the conditions:

U1. If G ε then G has a finite series

with locally nilpotent factors.

Serial subgroups seem as yet to have received little attention in the literature. In this short note we draw attention to some interesting properties which these subgroups possess in locally finite groups.

The theory of dual spaces and duality is extended from a pair of vector spaces in duality to a list of more than two vector spaces in mutuality. The notions of a canonical bilinear functional and of compatible topologies for a dual pair are extended to those of a canonical multilinear functional and of compatible topologies for a mutual list. Compatible topologies are characterized by means of an extended version of the Mackey–Arens Theorem. Directions for further work are noted.

From the tensorial point of view, the essential characteristic of a semisimple lie algebra A is the non-singularity of the Killing form, since this enables one to construct a system of Cartesian tensors over A. That system is the subject of this paper.

One of the most studied parameters in all of graph theory is the chromatic number. Undoubtedly, its popularity as a subject for research is due to its intimate relationship with the famous Four Colour Problem.

Notation: Let M and Q be manifolds. The embedding f: M → Q is said to be proper if f−1∂Q = ∂M.

1. The classical Kneser conjecture states that if M is a closed 3-manifold and π1(M) ≈ A * B, a free product, then there exist 3-manifolds MA, MB such that

and π1(MA) ≈ A, π1(MB) ≈ B. This has been proved by John Stallings(6).

It is the object of this paper to prove a transversality theorem for homology manifolds which is strong enough to be then able to prove a Thom theorem for the cobordism ring of homology manifolds. The new bundle theory necessary for this follows on naturally from that developed in ‘Homology Cobordism Bundles’(2).

This paper is concerned with the symplectic bordism ring MSp*, whose structure is still largely unknown despite the partial results of Liulevicius(3), Novikov(4) and Stong(lO). It is the first of two elaborating(8), and it relies completely on (7) for notation and prerequisites. In particular we assume the reader to be familiar with (7), section 8, which establishes the relation between the Sp Hattori-Stong conjecture (that KO decides MSp) and the spectral sequence

We are here concerned entirely with computations in this spectral sequence.

This paper is designed to pave the way for the author's work on the symplectic bordism ring MSp*(12), (13). We here discuss all the notation used, and collect together all the theorems quoted therein. At the same time, I hope that some of the results presented here might be of interest in their own right. Our central theme is the study of the hurewicz map e: S*(X) → E*(X), both in general and certain specific cases.

This is the second of two papers elaborating (4), and is concerned with the more geometrical aspects of the symplectic bordism ring in low dimensions. As such, it is a natural sequel to (2). However, the reader who does not care for the algebra therein need only consult (3), in which appears all the notation and prerequisites for this paper.

In (6) we studied a topology T on the product set X × Y of two topological spaces X and Y which was defined by the requirement that each mapping from X × Y which was continuous in each variable separately was also continuous in T; we called (X × Y, T) the tensor product of X and Y, and denoted it by X ⊗ Y. Theorem (3·2) of (6) indicated that X ⊗ Y was rarely completely regular; as complete regularity is of importance in analytic problems, we consider here a ‘completely regular tensor product’ . Roughly speaking, gives a tensor product in the category of completely regular topological spaces. The categorical properties of are discussed in section 5.

Let (E, C) be a partially ordered vector space with positive cone C. The order-bound topology Pb(6) (order topology in the terminology of Schaefer(9)) on E is the finest locally convex topology for which every order-bounded subset of E is topologically bounded.

We let G denote an infinite compact group and G its dual. We use the notation of our book ((l), Chapters 7 and 8). Recall A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and ℒ∞(G) denotes its dual space under the pairing 〈ƒ,φ〉 (ƒ ∈ A(G), φ ∈ ℒ∞(G)). Further, note ℒ∞(G) is identified with the C*-algebra of bounded operators on L2(G) commuting with left translation. The module action of A(G) of ℒ∞(G) is defined by the following: for ƒ ∈ A(G), φ ℒ∞(G), ƒ. φ ∈ ℒ∞(G) by 〈g, ƒ . φ〉 = 〈 ƒg, φ〉, g ∈ A(G) Also ‖ƒ . φ‖∞ ≥ ‖ƒ‖A ‖φ‖∞.

It is shown that for the Cesàro means (C, α) with α > - 1, and for a certain class of more general Nörlund means, summability of the series σan implies uniform summability of the series σan zn in a Stolz angle at z = 1.

If B is a normal matrix and (B) denotes the series summability field with the usual Banach space topology, then the vectors {ek} (ek = {0,0,..., 1,0,...}) are said to form a Toplitz basis for (B) relative to a method H if H — Σakek = a for each a = {ak}ε(B). It is shown for example that the above relation holds for B = (C,α), α> − 1 , and H = Abel method; also for B = (C,α) and H = (C,β) with 0 ≤ α ≤ β.

Applications are made to theorems on summability factors.

1. Let g(n) = g(n; m) denote the number of partitions of a natural number n into powers of a given natural number m (> 2). Then the generating function of g(n) is

(1) where we take g(0) = 1.

Introduction: Let X1, X2, …, Xn be n (n ≤ 2) independent observations on a random variable X with distribution function F. Also let L = L (X1, X2, …, Xn) be a linear statistic and Q = Q (X1, X2, …, Xn) be a homogeneous quadratic statistic. In this paper, we consider the problem of characterizing a class of probability distributions by the linear regression of the statistic Q on the other statistic L. In section 2, we obtain a characterization of a class of probability distributions, which includes the normal and the Poisson distributions. In section 3, a class of distributions including the gamma, the binomial and the negative binomial distributions is characterized.

The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.