A negative answer to the Kuroš–Černikov Question 21 in [7], whether a group satisfying the normalizer condition is hypercentral, was given by Heineken and Mohamed in 1968 [6]. They constructed groups G satisfying:

(i) G is a locally finite p-group for a prime p,

(ii) G/G′≅Cp∞ and G′ is countable elementary abelian,

(iii) every proper subgroup of G is subnormal and nilpotent,

(iv) Z(G)={1},

(v) the set of normal subgroups of G contained in G′ is linearly ordered by set inclusion, see [3, p. 334],

(vi) KG′ is a proper subgroup in G for every proper subgroup K of G, see [6, Lemma 1(a)].

Inspired by the work of Bloch and Kato in [2], David Burns constructed several ‘equivariant Tamagawa invariants’ associated to motives of number fields. These invariants lie in relative $K$-groups of group-rings of Galois groups, and in [3] Burns gave several conjectures (see Conjecture 3.1) about their values. In this paper I shall verify Burns' conjecture concerning the invariant $T\Omega^{\rm loc}( N/{\bf Q},1)$ for some families of quaternion extensions $N/{\bf Q}$. Using the results of [9] I intend in a subsequent paper to verify Burns' conjecture for those families of quaternion fields which are not covered here.

We introduce the concept of ‘geometrical spine’ for 3-manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of $L_{p,q}$ that is close enough to its geometrical spine contains at least $E(p,q)-3$ vertices, which is exactly the conjectured value for the complexity $c(L_{p,q})$. As a byproduct, we find the minimal rotation distance (in the Sleator–Tarjan–Thurston sense) between a triangulation of a regular $p$-gon and its image under rotation.

Standard codeterminants for Donkin's symplectic Schur algebra $\bar{S}(n,r)$ are defined. It is shown that they form a basis of $\bar{S}(n,r)$. They are then used to give a purely combinatorial proof that $\bar{S}(n,r)$ is quasi-hereditary.

In this paper, we prove the following theorems. (i) Let G be a graph of minimum degree δ[ges ]5. If G is embeddable in a surface σ and satisfies (δ−5)[mid ]V(G)[mid ]+6χ(σ)[ges ]0, then G is edge reconstructible. (ii) Any graph of minimum degree 4 that triangulates a surface is edge reconstructible. (iii) Any graph which triangulates a surface of characteristic χ[ges ]0 is edge reconstructible.

It is shown that many hyperbolic monopoles can be distinguished from each other via their asymptotic values in contrast to the case of Euclidean monopoles.

The Beurling algebras $l^1({\cal D},\omega)\;({\cal D}={\bb N},{\bb Z})$ that are semi-simple, with compact Gelfand transform, are considered. The paper gives a necessary and sufficient condition (on $\omega$ ) such that $l^1({\cal D},\omega)$ possesses a uniform quantitative version of Wiener's theorem in the sense that there exists a function $\phi:]0,+\infty[\longrightarrow ]0,+\infty[$ such that, for every invertible element $x$ in the unit ball of $l^1({\cal D},\omega)$ , we have \[ \|x^{-1}\|\le \phi(r(x^{-1}))\quad r(x^{-1})\hbox{ is the spectral radius of }x^{-1}. \]

Let T = {T(t)}t[ges ]0 be a C0-semigroup on a Banach space X. The following results are proved.

(i) If X is separable, there exist separable Hilbert spaces X0 and X1, continuous dense embeddings j0[ratio ]X0 → X and j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that j0 ∘ T0(t) = T(t) ∘ j0 and T1(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0.

(ii) If T is [odot ]-reflexive, there exist reflexive Banach spaces X0 and X1 , continuous dense embeddings j[ratio ]D(A2) → X0, j0[ratio ]X0 → X, j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that T0(t) ∘ j = j ∘ T(t), j0 ∘ T0(t) = T(t) ∘ j0 and T(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0, and such that σ(A0) = σ(A) = σ(A1), where Ak is the generator of Tk, k = 0, [emptyv ], 1.

It is proved that there are two minimal $\Pi_1^0$ classes that are not automorphic.

All finite groups G are determined that admit a subgroup K of index pa, p a prime, under the assumption that G has an irreducible and faithful GF(q)-module in characteristic p whose dimension over GF(q) is at most a. As an application to the theory of permutation groups, the maximal transitive subgroups of the primitive affine permutation groups are determined. The above-mentioned classification is generalized by dropping the assumption that p|q. In both cases surprisingly nice results are obtained.

The first example of a non-finitely based variety of centre-by-abelian-by-nilpotent groups was found in 1995. This example has an infinite exponent but it can be seen that the method used to construct it also gives a variety that has an exponent of 128. Later it was proved that there exists a variety with similar properties but with an exponent of 32. The paper constructs a non-finitely based variety of centre-by-abelian-by-nilpotent groups which has an exponent of 8.

Let f be analytic and univalent in Δ={z:[mid ]z[mid ]<1}. We write

formula here

for 0[les ]r<1 and θ∈[0, 2π], and remark that L(r, θ) is the length of the image in f(Δ) of the ray joining 0 to reiθ in Δ. We shall be concerned with the growth of L(r, θ), as a function of r, for univalent functions in Δ and, more particularly, for functions in the subclass of close-to-convex univalent functions in Δ.

The fine topology on ℝn (n[ges ]2) is the coarsest topology for which all superharmonic functions on ℝn are continuous. We refer to Doob [11, 1.XI] for its basic properties and its relationship to the notion of thinness. This paper presents several theorems relating the fine topology to limits of functions along parallel lines. (Results of this nature for the minimal fine topology have been given by Doob – see [10, Theorem 3.1] or [11, 1.XII.23] – and the second author [15].) In particular, we will establish improvements and generalizations of results of Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihl and Seidel [6], Schneider [21], Berman [7], and Armitage and Nelson [4], and will also solve a problem posed by the latter authors.

An early version of our first result is due to Evans [12, p. 234], who proved that, if u is a superharmonic function on ℝ3, then there is a set E⊆ℝ2×{0}, of two-dimensional measure 0, such that u(x, y,·) is continuous on ℝ whenever (x, y, 0)∉E. We denote a typical point of ℝn by X=(X′ x), where X′∈ℝn−1 and x∈ℝ. Let π:ℝn→ℝn−1×{0} denote the projection map given by π(X′, x) = (X′, 0). For any function f:ℝn→[−∞, +∞] and point X we define the vertical and fine cluster sets of f at X respectively by

formula here

and

formula here

Sets which are open in the fine topology will be called finely open, and functions which are continuous with respect to the fine topology will be called finely continuous. Corollary 1(ii) below is an improvement of Evans' result.

It is shown that for certain discrete p-perfect groups G, in particular for all p-perfect finite groups and certain groups of finite virtual cohomological dimension, the loop space on the p-completed classifying space ΩBG∧p is a retract of the loop space on the p-completion of a certain finite complex. For those finite virtual cohomological dimension groups where the author's methods apply, a bound on the dimension of such a complex is provided.

The paper is concerned with rings of polynomial invariants of finite groups. In particular, it will be shown that these rings are isomorphic as modules over the Steenrod algebra [Pscr ]* if and only if the group representations are pointwise conjugate. An application to cohomology is the construction of classifying spaces of finite groups which are not homotopy equivalent, but where the cohomology rings are isomorphic as unstable modules over the (topological) Steenrod algebra.

In this paper, we study some asymptotic aspects of the positive eigenfunctions of the combinatorial Laplacian associated to a homogeneous tree. The results are inspired by results of Dennis Sullivan concerning λ-harmonic functions on the hyperbolic spaces ℍn and contained in the paper [9].

A new proof is given of a theorem of Menšov which states that a measurable function can be changed on a set of arbitrarily small measure to obtain a function with uniformly convergent Fourier series. Other results of Menšov on the representation of functions by trigonometric series are obtained by using a related result.

The paper is concerned with the uniqueness and existence problem for a multidimensional reacting and convection system with the vanishing viscosity method. The uniqueness theorem is obtained from the stability with respect to the initial data. To solve the existence problem, the uniqueness and existence of solutions to a viscous system are first proved, and then the L1-modulus of continuity of solutions independent of the small viscosity is obtained.

We show that given an affine algebraic group G over a field K and a finite subgroup scheme H of G there exists a finite dimensional G-module V such that V[mid ]H is free. The problem is raised in the recent paper by Kuzucuo˘glu and Zalesskiiˇ [15] which contains a treatment of the special case in which K is the algebraic closure of a finite field and H is reduced. Our treatment is divided into two parts, according to whether K has zero or positive characteristic. The essence of the characteristic 0 case is a proof that, for given n, there exists a polynomial GLn(ℚ)-module V of dimension 2nΠpp(n2), where the product is over all primes less than or equal to n+1, such that V is free as a ℚH-module for every finite subgroup H of GLn(ℚ). The module V is the tensor product of the exterior algebra Λ*(E), on the natural GLn(ℚ)-module E, and Steinberg modules Stp, one for each prime not exceeding n+1. The Steinberg modules also play the major role in the case in which K has characteristic p>0 and the key point in our treatment is to show that for a finite subgroup scheme H of a general linear group scheme (or universal Chevalley group scheme) G over K, the Steinberg module Stpn for G is injective (and projective) on restriction to H for n[Gt ]0. A curious consequence of this is that, despite the wild behaviour of the modular representation theory of finite groups, one has the following. Let H be a finite group and V a finite dimensional vector space. Then there exists a (well-understood) faithful rational representation π: GL(V)→GL(W) such that, for each faithful representation ρ: H→GL(V), the composite πορ: H→GL(W) is free, in particular all representations πορ are equivalent.

The close relationship between the notions of positive forms and representations for a C*-algebra A is one of the most basic facts in the subject. In particular the weak containment of representations is well understood in terms of positive forms: given a representation π of A in a Hilbert space H and a positive form φ on A, its associated representation π φ is weakly contained in π (that is, ker π φ ⊃ ker π) if and only if φ belongs to the weak* closure of the cone of all finite sums of coefficients of π. Among the results on the subject, let us recall the following ones. Suppose that A is concretely represented in H. Then every positive form φ on A is the weak* limit of forms of the type x [map ] [sum ]ki=1 〈ξi, xξi〉 with the ξi in H; moreover if A is a von Neumann subalgebra of ℒ(H) and φ is normal, there exists a sequence (ξi)i [ges ] 1 in H such that φ (x) = [sum ]i [ges ] 1 〈ξi, xξi〉 for all x.