Automorphism towers were first considered in (19) with the case of finite groups. Recently polycyclic and extremal groups were considered in (7) and (12) respectively. The purpose of this paper is to consider automorphism towers of groups in a class of almost soluble groups which contains all polycyclic and all extremal groups.

Although this paper is concerned with answering a question in general topology about sequential convergence, the techniques used may be of greater interest to those interested in the structure of filters on ω. A point in a topological space is called an α1-point if whenever is a countable family of sequences converging to it, there is a sequence B, also converging to it, such that A\B is finite for each A ∈ . A space is α1 if each point of the space is an α1-point. Nyikos has shown that the existence of countable Fréchet α1-spaces which are not first countable follows from the assumption and has asked if it is consistent that no such space exists. It is not difficult to see that if there is an α1-space which is not first countable the there is also one with only one non-isolated point. Since, in answering Nyikos' question, we are only concerned with countable spaces, it follows that we are really dealing with certain types of filters on ω. The precise translation of the topological question to a filter theoretic one is contained in §2. The reader who is not interested in the details may wish to read only §2, the first half of §3 and §5.

The object of this paper is to make more precise the results given by Enriques-Campedelli for canonical surfaces, that is, surfaces whose prime sections are the canonical curves. We consider the case of regular surfaces in space of three dimensions (pg = pn = 4), and of regular multiple planes (pg = pn = 3).

According to the result by Dyer and Formanek [4], the automorphism group of a finitely generated free two-step nilpotent group is complete except in the case when this group is a one- or three-generator (the three-generator groups have automorphism tower of height 2). The purpose of this paper is to prove that the automorphism group of an infinitely generated free two-step nilpotent group is also complete. (Recall that a group G is said to be complete if G is centreless and every automorphism is inner.)

The paper may be considered as a contribution to the study of automorphism towers of relatively free groups.

The object of this paper is to prove a necessary and sufficient condition on two groups H, K for their join always to be subnormal in a group G whenever they are embedded subnormally in G.

A scheme is presented to show the limiting behaviour of the electronic energy levels in the diatomic one-electron problem for unequal centres, as the ions are separated. This generalizes the well known order of levels in the hydrogen molecule ion. Certain implications of the result for the properties of asymmetric diatomic molecules are briefly discussed.

Let K be a cohomology theory. Axioms are given which uniquely characterize the transfer in K-cohomology for bundles with structure group a compact Lie group and fibre a finite C.W. complex. Also, a family of generalized transfers is defined which includes both the standard transfer and Atiyah's holomorphic transfer.

0. Introduction. In a paper (1) with the same title, one of us attempted to classify the reducible k2-bundles on a quadric surface Q = P1 × P1 defined over an algebraically closed field k. This attempt suffered from two defects: first, the classification was given by a one-one correspondence rather than by an algebraic parameter variety, and, secondly, there is a mistake in the proof of Proposition 6 of (1), as a result of which Proposition 6, Proposition 7 and the exceptional clause of the Theorem in (1) are false. To correct these defects we use the definitions, notations and numbering of (1) without comment. Finally, we show how the classification of reducible k2-bundles on P1 × … × P1 (n factors) is determined by that of reducible k2-bundles on P1 × P1. The net effect of these corrections is to underline the moral of (1): while the classification of reducible k2-bundles (but with a distinguished factor of the product variety P1 × … × P1) is as nice as it could possibly be, the classification of reducible k2-bundles alone is as nasty as it could possibly be. Since every decomposable k2-bundle is just a sum of two line bundles, we consider only indecomposable reducible k2-bundles.

1. In this paper, the author derives a formula of inversion for the integral transform

where a > 0, k > 0. The actual formula obtained appears as equation (8) of the present paper and is deduced from an integral formula constructed in (1).

This paper is a continuation of [15]. In that paper, I introduced a general framework which allows one to produce ‘weak’ counterexamples to Torelli for Calabi–Yau threefolds: deformation families containing non-isomorphic varieties Yt, Y+t with isomorphic Hodge theory on the third cohomology. The varieties arise as deformations of threefolds Y that are resolutions of singular varieties X with rather special properties (cf. Section 1). In [15], I discussed two families containing suitable X that do provide a counterexample and a third family with remarkably similar properties where however the existence of a nontrivial generic automorphism destroys the counterexample. This shows that explicit examples are necessary; there is a precise set of conditions one needs to check carefully in order to obtain counterexamples to Torelli (Theorem 1·1). In fact, there exist several families with renitent automorphisms (Remark 4·5).

Using the concept of $s$-formality we are able to extend the bounds of a Theorem of Miller and show that a compact $k$-connected $(4k+3)$- or $(4k+4)$-manifold with $b_{k+1}=1$ is formal. We study $k$-connected $n$-manifolds, $n=4k+3, 4k+4$, with a hard Lefschetz-like property and prove that in this case if $b_{k+1}=2$, then the manifold is formal, while, in $4k+3$-dimensions, if $b_{k+1}=3$ all Massey products vanish. We finish with examples inspired by symplectic geometry and manifolds with special holonomy.

Let K be a field and let R be a regular domain containing K. Let G be a finite subgroup of the group of automorphisms of R. We assume that |G| is invertible in K. Let RG be the ring of invariants of G. Let I be an ideal in RG. Fix i ⩾ 0. If RG is Gorenstein then:

1. (i) injdimRGHiI(RG) ⩽ dim Supp HiI(RG);

2. (ii) $H^j_{\mathfrak{m}}$(HiI(RG)) is injective, where $\mathfrak{m}$ is any maximal ideal of RG;

3. (iii) μj(P, HiI(RG)) = μj(P′, HiIR(R)) where P′ is any prime in R lying above P.

In this paper, we give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type Nn ×f Qq, where f ∈ C∞(N). This setting allows us to deal, among other things, with minimal submanifolds bounded by cylinders, cones, spheres and pseudo-hyperbolic spaces where most of these examples are not covered in the literature. Applications also include the study of the essential spectrum of hyperbolic graphs over compact regions of the boundary at infinity.

C. A. Rogers and J. E. Jayne have asked whether, given a Polish space and an analytic subset A of which is not a Borel set, there is always a compact subset K of such that, A ∩ K is not Borel. In this paper we give both a proof, using Martin's axiom and the negation of the continuum hypothesis, of and a counter-example, using the axiom of constructibility, to the conjecture of Rogers and Jayne, which set theory with the axiom of choice is thus powerless to decide.

Let the generalized hypergeometric function of one variable be denoted by

where (a)m is the Pochhammer symbol ((1, 3)).

Let $\mathbb{K}$ be a number field, Γ a finitely generated subgroup of $\mathbb{K}$*, for instance the unit group of $\mathbb{K}$, and κ>0. For an ideal $\mathfrak{a}$ of $\mathbb{K}$ let indΓ($\mathfrak{a}$]></alt-text></inline-graphic>) denote the multiplicative index of the reduction of Γ in <inline-graphic name="S0305004114000206_inline3"><alt-text><![CDATA[$(\mathcal{O}_\mathbb{K}/\mathfrak{a})$* (whenever it makes sense). For a prime ideal $\mathfrak{p}$ of $\mathbb{K}$ and a positive integer γ let $\mathcal{I}_\gamma^\kappa(\mathfrak{p})$ be the average of ${ind}_{\langle a_1,\dots,a_\gamma\rangle}(\mathfrak{p})^\kappa$ over all tupels $(a_1,\dots,a_\gamma)\in{(\mathcal{O}_\mathbb{K}/\mathfrak{p})^*}^\gamma$. Motivated by a problem of Rohrlich we prove, partly conditionally on fairly standard hypotheses, lower bounds for $\sum_{\mathcal{N}{\mathfrak{a}\leq x}{ind}_{\Gamma}({\mathfrak{a})^\kappa$ and asymptotic formulae for $\sum_{\mathcal{N}\mathfrak{p} \leq x} {\mathcal{I}_{\gamma}^\kappa({\mathfrak{p})$.

The projective representations of a finite group G over a field K are divided into sets, each parametrized by an element of the group H2(G, Kx). The latter is the Schur multiplier M(G) when K = ℂ.

For G a classical group of type Bn, Cn or Dn defined over an algebraically closed field (of characteristic other than two if G is a special orthogonal group), we show that the centralizer algebra of G on the space of endomorphisms of the module of traceless tensors Wr is naturally isomorphic to the group algebra of the symmetric group Sr if r[les ]n (for G of type Bn or Cn) or if r<n (for G of type Dn). This extends a characteristic zero result of Brauer and Weyl and recovers some of the related characteristic free results that De Concini and Strickland obtained for G of type Cn.

Let G be a simple algebraic group over an algebraically closed field K of characteristic p. If Σ is the root system of G and Uα is the root subgroup of G corresponding to a long root α∈Σ, then 〈Uα, U−α〉 is an image of SL2 and any G-conjugate of this subgroup is called a fundamental subgroup of G. In [LS1], the closed connected semisimple subgroups of G generated by long root elements were determined. Of course, long root elements are unipotent elements of fundamental subgroups. In this paper we consider subgroups of G which are generated by semisimple elements lying in fundamental subgroups.

Our first three results (Theorems 1–3) concern semisimple connected subgroups of G which contain a maximal torus of a fundamental subgroup; we call such a torus a fundamental torus. Notice that for classical groups, the elements in fundamental tori have fixed spaces of small codimension in the natural module; indeed, for the groups SL(V) and Sp(V), the elements of fundamental tori are precisely those semisimple elements with fixed space of codimension 2, while for SO(V), the codimension is 4.

As consequences of these results, we obtain information on subgroups (finite or infinite) of classical groups which are generated by conjugates of a single element of a fundamental torus of order at least 5 (see Theorems 4, 5 and Corollary 6); for example, Theorem 5 determines those finite irreducible subgroups containing such an element which are quasisimple and of Lie type in characteristic p.

We now state our results in detail. Recall from [LS1] that a subsystem subgroup of G is a connected semisimple subgroup which is invariant under a maximal torus of G.