Let the finite group A be acting on a finite group G with ([mid ]A[mid ], [mid ]G[mid ])=1. Let Γ be the semidirect product of A and G. Let χ be a character of Γ irreducible after restriction to G. In a previous paper by Brian Hartley and the author, we proved that the restriction of χ to S belongs to the set [Cscr ](S) obtained by running over all χ that arise in this manner, by assuming, in addition, that G is a product of extraspecial groups. This was proved in general, assuming only some condition on the Green functions of groups of Lie type that is not as yet fully verified. In the present paper, we define the map Q(χ): S[map ][Copf ] by Q(χ)(s) =[mid ]CG(s)[mid ]/χ(s). We prove that Q(χ)∈[Cscr ](S) under the same hypotheses. In particular, the character quotient Q(χ) is an ordinary character.

JB*-triples occur in the study of bounded symmetric domains in several complex variables and in the study of contractive projections on C*-algebras. These spaces are equipped with a ternary product {·,·,·}, the Jordan triple product, and are essentially geometric objects in that the linear isometries between them are exactly the linear bijections preserving the Jordan triple product (cf. [23]).

We show that solutions of analytic elliptic partial differential equations of the form

formula here

in a simply connected domain Ω ⊂ ℝn can be extended holomorphically into [Nscr ] (Ω) ⊂ [Copf ]n, the so-called kernel of the harmonicity hull of Ω. This extends results of Avanissian, Aronszajn etc., on polyharmonic functions and also results of Vekua, Khavinson and Shapiro in ℝ2. We also find the domain of influence for solutions of a certain subclass of these operators, in terms of their Cauchy data on analytic hypersurfaces in [Copf ]n (a complex Huygens' principle). As an application, we investigate reflection properties of these solutions and, in particular, solutions of the Helmholtz equation in ℝ3.

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.

Certain projections of the real polytopes {3, 3, 4}, {3, 4, 3}, {3, 3, 5} suggest highly symmetric coordinates for the self-reciprocal complex polygons 3{3}3, 4{3}4, 3{4}3, 5{3}5 and 3{5}3. Although there are a number of interesting complications, this suggestion is essentially correct and leads to elegant coordinates for all the sporadic complex polygons. Among the by-products of producing these coordinates we count most significant our new insights about 2{6}3 and our simple proof that the 600 vertices of the real polytope {5, 3, 3} are quite unrelated to the 600 vertices of either 5{6}2 or 5{4}3.

A stage in the classification of the periodic simple finitary linear groups by J. I. Hall and others, is a partial reduction to the countable case; specifically a simple group is isomorphic to a finitary linear group if and only if its countable subgroups are isomorphic to finitary linear groups. It is natural to ask, how common is it for a group to have a faithful finitary linear (or possibly only skew linear) representation, whenever each of its countable subgroups has such a representation? In very recent work F. Leinen and O. Puglisi have shown that periodic primitive finitary linear groups are countably recognizable, see [4]. It is easy to see that every abelian group is isomorphic to a finitary linear group, for example over the subfield of the complexes [Copf ] generated by all roots of unity. Thus the first non-obvious case, at the opposite extreme to the simple groups (and also in some sense to the primitive groups), is that of the nilpotent groups. We show the following, which, it should be noted, covers the case of periodic nilpotent groups.

A large number of papers written over the last ten years have concerned the spectral theory of Laplace–Beltrami operators on complete Riemannian manifolds, and of other self-adjoint second order elliptic operators. Much of the interest has centred on the relationship between various types of Sobolev inequality, parabolic Harnack inequalities and the Liouville property on the one hand, and Gaussian heat kernel bounds on the other. For manifolds of bounded geometry there is an important connection between this problem and a corresponding one for discrete Laplacians on graphs. Standard references are [9, 37] and more recent literature can be traced via [5, 16, 32].

We construct a one-parameter family of sets in ℝ2 generated by a disjoint union of open squares. We study the spectral counting function associated to a variational Dirichlet eigenvalue problem on a set from this family and show that the spectral asymptotics depend not only on the Minkowski dimension of the boundary, but also on whether the values of a specific function of the parameter are rational or irrational. Furthermore, we significantly sharpen the results in the rational case.

In the nineteenth century, field theory brilliantly resolved a number of questions that had taxed mathematicians for centuries; for example, ‘The circle cannot be squared’ by straight edge and compass, and solving polynomial equations by radicals is not always possible. These successes have continued to be held up as superb examples of the power of mathematical thought, and are demonstrated at an undergraduate level. The purpose of this article is to provide another such natural example which leads to a concrete realisation of the free group on 2 generators.

For reciprocation with respect to a sphere [sum ]x2=c in Euclidean n-space, there is a unitary analogue: Hermitian reciprocation with respect to an antisphere [sum ]uu=c. This is now applied, for the first time, to complex polytopes.

When a regular polytope Π has a palindromic Schläfli symbol, it is self-reciprocal in the sense that its reciprocal Π′, with respect to a suitable concentric sphere or antisphere, is congruent to Π. The present article reveals that Π and Π′ usually have together the same vertices as a third polytope Π+ and the same facet-hyperplanes as a fourth polytope Π− (where Π+ and Π− are again regular), so as to form a ‘compound’, Π+[2Π]Π−. When the geometry is real, Π+ is the convex hull of Π and Π′, while Π− is their common content or ‘core’. For instance, when Π is a regular p-gon {p}, the compound is

formula here

The exceptions are of two kinds. In one, Π+ and Π− are not regular. The actual cases are when Π is an n-simplex {3, 3, ..., 3} with n[ges ]4 or the real 4-dimensional 24-cell {3, 4, 3}=2{3}2{4}2{3}2 or the complex 4-dimensional Witting polytope 3{3}3{3}3{3}3. The other kind of exception arises when the vertices of Π are the poles of its own facet-hyperplanes, so that Π, Π′, Π+ and Π− all coincide. Then Π is said to be strongly self-reciprocal.

We characterise the profinite groups with polynomial subgroup growth. We deduce that the property PSG is extension-closed in the category of all groups, and subgroup-closed in the category of finitely generated profinite groups.

We give a necessary and sufficient condition on an elliptic symbol σ of order m to ensure that the unique closed extension in Lp(ℝn) for 1 < p < ∞, of the pseudo-differential operator Tσ, initially defined on the Schwartz space, is a Fredholm operator from Lp(ℝn) into Lp(ℝn) with domain Hm, p, where Hm, p is the Lp Sobolev space of order m.

The Bers–Greenberg theorem tells us that the Teichmüller space of a Riemann surface with branch points (orbifold) depends only on the genus and the number of special points, and not on the particular ramification values. On the other hand, the Maskit embedding provides a mapping from the Teichmüller space of an orbifold, into the product of one-dimensional Teichmüller spaces. In this paper we prove that there is a set of isomorphisms between one-dimensional Teichmüller spaces that, when restricted to the image of the Teichmüller space of an orbifold under the Maskit embedding, provides the Bers–Greenberg isomorphism.

If F is a subset of G⊆GL (n, [Kscr ])=GL(V, [Kscr ]) (where [Kscr ] is a field) the degree of F(=deg (F)) is the dimension of the [Kscr ]-space [V, F] spanned by

{v(f−1)[mid ]v∈V, f∈〈F〉};

note that in the special case F={g} we have [V, g]={v(g−1)[mid ]v∈V}. Our intention is to describe those irreducible linear groups G⊆GL (n, [Kscr ]) generated by elements whose degrees are small relative to n. To do this successfully it seems necessary to work within the restricted class of ‘solvable-by-locally finite’ groups (throughout, this class will be denoted by [Sscr ](L[Fscr ])). Somewhat surprisingly, it turns out that if G is an irreducible [Sscr ](L[Fscr ])-subgroup of GL (n, [Kscr ]) generated by elements of small degree (relative to n), then G has large non-abelian simple sections. For a linear group G, the restriction to the class [Sscr ](L[Fscr ]) is equivalent to insisting that G have no non-cyclic free subgroups (see [7, Section 5.6]). Our main result in this direction is the following structure theorem.

We study the boundedness of the Hardy–Littlewood maximal function in weighted Lorentz spaces, using some new techniques on distribution and rearrangement functions. Our goal is to give a unified version of some well-known weighted inequalities in ℝn and ℝ+, showing the relation between the theories.

The first examples of triangular AF algebras to be studied were the refinement algebra and the standard algebra. Both are analytic algebras with the property that the cocycle can be taken to be constant on the matrix units of the algebra. The latter property, called local constancy, is quite special and is still ill-understood. In the present paper we examine the class of nest algebras [Tscr ] in AF C*-algebras which share the distinctive features of the refinement algebra:

(1) [Tscr ] is a nest algebra in which the nest generates the diagonal,

(2) [Tscr ] admits a locally constant cocycle.

It is well known that the product of normal locally nilpotent (locally polycyclic) groups is again locally nilpotent (locally polycyclic). This is not true for locally solvable groups. We look at a class between locally polycyclic and locally solvable groups that is closed under normal products.

In order to obtain the linear relations among the 24-solutions for the p-differential equation of Riemann, Barnes [12] discovered an elegant method which uses the following identity for the classical Γ function.

For complex numbers α, β, γ, δ the equality

formula here

holds and the path of integration from −i∞ to +i∞ is curved so that the poles of Γ(γ−s) Γ(δ−s) lie on the right and the poles of Γ(α+s) Γ(β+s) lie on the left.

To each finitely generated nilpotent group Γ, we associate an integer d and show that the 2-dimensional isoperimetric function (the Dehn function) of Γ is bounded below by a polynomial function of degree d. In the case of free nilpotent groups of class c we find (if we exclude ℤ) that d=c+1.

Let X and Y be pointed spaces. A phantom map from X to Y is a map whose restriction to any finite skeleton of X is null-homotopic. Let Ph (X, Y) denote the set of homotopy classes of phantom maps from X to Y. As a pointed set it is isomorphic to the lim1 term of the tower of groups

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where Y(n) denotes the Postnikov approximation of Y through dimension n. The homomorphisms in this tower are induced by the projections ΩY(n)← ΩY(n+1)). The groups in this tower are not abelian in general; however they do have some nice algebraic properties.