An n-link of multiplicity is a smooth embedding of the disjoint union of μ copies of Sn in Sn+2; is said to be trivial if it extends to a smooth embedding of the disjoint union of μ copies of Dn+1. Let , and Cnμ denote the wedge product of μ copies of S1 and (μ – 1) copies of Sn+1. Then clearly, if is trivial, then X ≃ Cn,μ, where ≃ denotes homotopy equivalence.

A claim of H. H. Teh on Archimedean full orders ((2), ‘only if’ of Theorem 2, p. 480) fails. His statement (p. 478, line 5) that the direct product of n copies of the additive rational numbers admits no full order of Archimedean rank more than n is proved without Hahn's embedding theorem. A few other remarks are made.

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)).

Let X be a closed, compact connected 2-manifold (a surface), which we will denote by O or N if we wish to stress that X is orientable or non-orientable. Let G(X) denote the group of all homeomorphisms X → X, D(X) the normal subgroup of homeomorphisms isotopic to the identity, and H(X) the factor group G(X)/D(X), i.e. the homeotopy group of X. The problem of determining generators for H(O) was considered by Lickorish in (7, 8), and the second of these papers specifies a finite set of generators of a particularly simple type. In (10) and (11) Lickorish considered the analogous problem for non-orientable surfaces, and, using Lickorish's partial results, Chilling-worth (4) determined a finite set of generators for H(N). While the generators obtained for H(O) and H(N) were strikingly similar, it was noteworthy that the techniques used in the two cases were different, and in particular that little use was made in the non-orientable case of the earlier results obtained on the orientable case. The purpose of this note is to show that the results of Lickorish and Chillingworth on non-orientable surfaces follow rather easily from the work in (7, 8) by an application of some ideas from the theory of covering spaces (2). Moreover, while Lickorish and Chillingworth sought only to find generators, we are able to show (Theorem 1) how in fact the entire structure of the group H(N) is determined by H(O), where O is an orientable double cover of N. Finally, we are able to determine defining relations for H(N) for the case where N is the connected sum of 3 projective planes (Theorem 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.

Notation and results. The main results of this paper are (a) a new proof of the canonical Schoenflies theorem and (b) a non-compact version of the canonical Schoen-flies theorem.

The result (a) is already known(5); the existing proof is based on the proof by Brown of the ordinary Schoenflies theorem, whereas ours is based on that of Mazur and Morse.

Throughout X will denote a connected, finite C.W. complex.

Let G be a subgroup of ∑n, the symmetric group, which acts transitively on the Cartesian product, Xn, of the space X. A map f:Xn→X is G-symmetric if it commutes with the action of G. If x0εX is a base point let i:X→Xn denote the inclusion, i(x) = (x,x0, …,x0). In ((6); (7); (11)–(13)) the following problem is posed: if X is an orientable topological n-manifold, what is the set of integers which may be obtained as the degree of(f.i) where f is a G-symmetric map? The degree of(f.i) is called the James number of f. If G = ∑n(G = Zn) a G-symmetric map will be called a symmetric map (a cyclic map). If X = Sn, the n-sphere, this problem has been studied in ((6)–(8), (11)–(13))

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.

Let Mm be a closed connected smooth manifold of dimension m, and set Pm = Mm – int Dm where Dm is a disc in M. In (4), Wall has the following exact sequence

where (M), (resp. (P)) is the Δ-set of diffeomorphisms of M (resp. P) as in(l), and [M/P] is the set of diffeomorphism classes of smooth manifolds obtained by glueing a disc to the boundary of P. In this paper we obtain some results on π0((M)) for particular M, and in the following sense: Using the techniques of (2), we can determine π0((P)), so the main portion of the paper is concerned with a discussion of the kernel of α. There is a map ω: π1((P)) → Γm+1, given by the obstruction to extending h∈π1((P)) to a concordance of the identity of M to itself, and it is clear that if this map is attached at the beginning of the sequence (*) we get exactness. We will obtain (for certain M) an alternative description of the image of α in terms of those homotopy spheres which can appear as the boundaries of thickenings.

An ideal of an integral group ring is divisible by a given integer if all of its elements share this common factor; the ideals most often encountered are rarely divisible in this sense. Only in the case of finite p-groups are powers of the augmentation ideal of the integral group ring ever divisible. For every e, there is some n for which the nth power of the augmentation ideal is divisible by pe. The smallest such integer n arises in many contexts; this paper describes its properties and interpretations.

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.

Atiyah(2) defined the geometrical dimension of an element to be less than k + 1 (g dim (x) ≤ k) if there is a k-dimensional bundle over X whose stable equivalence class is x. If ξ is an n-plane complex bundle over X, we say that it has r sections if there is an (n − r)-plane bundle η such that ξ is isomorphic to η ⊕ εr where εr is the trivial r-plane bundle over X. If X has dimension 2n or less, then ξ has r sections if and only if g dim (ξ − n) ≤ n − r.

Associated with, the powers of the augmentation ideal are the dimension subgroups. In the integral group ring case, they have long been conjectured to be the terms of the lower central series. This paper investigates the subgroups associated with the chain of ideals dual to the chain of powers of the augmentation ideal. The study is reduced to the case of the modular group rings of p-groups. The subgroups are calculated for Abelian p-groups, p odd. They appear in the upper central series of wreath products and provide a new criterion for the nilpotence of an arbitrary wreath product. The nilpotence class of wreath products is considered here as well; calculations and bounds are given; in particular, a new method of computing the class of the Sylow p-subgroups of the symmetric group arises.

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.

An ordered topological space is a set X endowed with a topology τ and a partial order ≤. We shall denote such a space by (X, τ), it being understood that (unless otherwise stated) the symbol ≤ is used to denote the partial order on X. An account of some of the properties of these spaces can be found in (5), (6) and (7).

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.

We consider some special operators on a locally convex Hausdorff space to itself, which have neat spectral theories and prove some perturbation results. This leads us to define and study a few special classes of locally convex spaces in which various subsets of the algebra of continuous linear operators either coincide or are closely related with each other. These are then compared to the classes of barrelled, infrabarrelled and DF-spaces and examples are given to distinguish them from one another.

We show that if E is a V-Sidon set and K a disjoint compact set then there exists a function of low tensor norm separating them.

Let p = (pk) and q = (qk) be any sequences of strictly positive numbers. Denote by c0(p) the set of all complex sequences x = (xk) such that . The Köthe–Toeplitz dual of c0(p) was determined in Theorem 6(4). In the case in which p∈l∞ (the space of all bounded sequences), c0(p) becomes a locally convex FK space under the paranorm where M = max (1, sup pk). It has been recently proved by Lascarides(3) that the following three properties are equivalent:

(a) p∈c0(i.e.pk → 0),

(b) c0(p) is perfect (in the sense of Köthe-Toeplitz duality),

(c) c0(p) has the Schur property (i.e. weak and strong sequential convergence are equivalent).

A function analytic approach to the generalized inversion problem, for arbitrary operators between Hilbert spaces, is investigated in the present paper.

The generalized inverse is defined as the inverse of the largest invertible restriction of the given operator and it is extended by zero to the orthogonal complement of the range of the invertible restriction. This domain-dense operator satisfies, on some restricted manifolds, the defining properties of the generalized inverse in some special cases, and it provides the least extremal solution of possibly inconsistent linear equations.