Denote by f(n) the number of subgroups of the symmetric group Sym(n) of degree n, and by ftrans(n) the number of its transitive subgroups. It was conjectured by Pyber [9] that almost all subgroups of Sym(n) are not transitive, that is, ftrans(n)/f(n) tends to 0 when n tends to infinity. It is still an open question whether or not this conjecture is true. The difficulty comes from the fact that, from many points of view, transitivity is not a really strong restriction on permutation groups, and there are too many transitive groups [9, Sections 3 and 4]. In this paper we solve the problem in the particular case of permutation groups of prime power degree, proving the following result.

It is known that quadratic polynomials do not have real rational orbits of period four. By using a two-dimensional model for the quadratic family, this result is generalized for complex rational orbits.

Holomorphic principal bundles over a compact Riemann surface X that admits a flat connection are considered. A holomorphic G-bundle over X, where G is a connected semisimple linear algebraic group over ${\Bbb C}$, admits a flat connection if and only if the adjoint vector bundle admits one. More generally, for a complex reductive group G, the necessary and sufficient condition on a G-bundle to admit a flat connection is described. This simplifies the criterion obtained by the authors and given in Math. Ann. 322 (2002) 333–346.

Nous donnons une majoration de la somme d'exponentielles SM = [sum ]Mm=1e(f(m)) lorsque la dérivée cinquième de f est d'un ordre de grandeur constant petit, noté λ, en fonction de M et de λ, améliorant un résultat ancien de Van Der Corput. La démonstration utilise un théorème de moyenne des puissances sixièmes de sommes d'exponentielles qui fait l'objet d'un article indépendant [6].

We give a bound for the exponential sum SM = [sum ]Mm=1e(f(m)) where f is a real-valued function whose fifth derivative is of a constant small size, say λ, by means of M and λ, improving an old result of Van Der Corput. The proof relies on a mean value theorem for sixth powers of exponential sums which is treated independently in [6].

Using the twistorial approach and some previous results, we prove the conjecture that the dimension of the moduli space of harmonic maps of area $4 \pi d$ from the 2-sphere to the $2n$-sphere is $2 d+n^2$ for the particular case $n=3$.

The existence of periodic solutions for the nonlinear asymmetric oscillator $ x''{+}\alpha x^+{-}\beta x^- {=} h(t),\qquad (' = d/dt) $ is discussed, where $\alpha, \beta$ are positive constants satisfying $ {1}/{\sqrt{\alpha}} + {1}/{\sqrt{\beta}} = {2}/{n} $ for some positive integer $n\in {\bf \mathbb{N}}$ and $h(t)\in L^\infty(0,2\pi)$ is $2\pi$-periodic with $x^{\pm}=\max\{\pm x, 0\}$.

Power series $\sum_{n=0}^\infty f(n)x^n$ with non-zero convergence radius $R(f)$ are considered, and the arithmetical nature (that is, irrationality, or even transcendence) of the corresponding multivariate series $\smash{\sum_{i_1,\ldots,i_m\!=0}^\infty f(i_1+\ldots +i_m)x_1^{i_1}\cdot\ldots \cdot x_m^{i_m}}$ is studied if $x_1,\ldots,x_m$ and the sequence $(f(n))$ satisfy appropriate arithmetical conditions. It follows that such arithmetical results can be written down easily if linear independence results on the function $F(x)$, defined in $|x|\,{<}\,R(f)$ by the original one-dimensional power series, and possibly its derivatives at the points $x_\mu$ are known. Some typical applications are explicitly stated.

Let $f \in \Z[x]$ be a polynomial of degree d. The paucity of non-trivial positive integer solutions to the equation $f(x_1)+f(x_2)=f(x_{3})+f(x_{4})$ is established, provided that $d \geq 7$. Also the corresponding situation is investigated for equal sums of three like polynomials.

For any positive integer n we let $P(n)$ be the largest prime factor of n. We improve and generalize several results of P. Erdős and C. Stewart on $P(n!+1)$. In particular, we show that $\limsup_{n \to \infty}P(n!+1)/n \ge 2.5$, which improves their lower bound of $\limsup_{n \to \infty} P(n!+1)/n >2$.

We study the Čerednik–Drinfeld p-adic uniformization of certain Atkin–Lehner quotients of Shimura curves over ℚ. We use it to determine over which local fields they have rational points and divisors of a given degree. Using a criterion of Poonen and Stoll, we show that the Shafarevich–Tate group of their jacobians is not of square order for infinitely many cases.

Let X be a separable infinite dimensional Banach space. There exist a closed set A ⊂ X which contains a translate of any compact set in the unit ball of X, and a bi-Lipschitz homeomorphism f of X onto X so that every line in X intersects f(A) in a set of Lebesgue measure zero.

This paper proves that a subgroup of finite index in a positively finitely generated profinite group has maximal subgroup growth at most $n^{\log (n\!)}$. In particular such a subgroup cannot be free, answering a question by L. Pyber.

In this paper, we show first that any prime (or semiprime) element of a commutative Banach algebra must have closed range. As a corollary, we find that in a commutative radical Banach algebra, all primes are zero divisors; indeed, all semiprimes are zero divisors (see below for the definition of semiprimeness). Our result is also true of a semiprime that is in the centre of a noncommutative Banach algebra.

The proof is fairly simple and entertaining, and we obtain a result that is helpful for the ambitious classification of elements in commutative radical Banach algebras being attempted by Marc Thomas. It is also related to the unbounded Kleinecke–Shirov conjecture.

We study finite centralizing extensions A⊂H of noetherian Hopf algebras. Our main results provide necessary and sufficient conditions for the fibres of the surjection spec H[Rarr ]spec A to coincide with the X-orbits in spec H, where X denotes the finite group of characters of H that restrict to the counit of A. In particular, all of the fibres are X-orbits if and only if the fibre over the augmentation ideal of A is an X-orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.

Let p be a prime number, and let G be a p-group which is not elementary abelian. For every mod-p cohomology class ξ of G which restricts trivially to all proper subgroups, we show that ξp = 0. This gives upper bounds for nilpotency degrees of such classes of G and of nilpotent mod-p cohomology classes of finite groups.

Let V be a finite-dimensional vector space over a finite field. The general and special linear groups, GL(V) and SL(V), act on the exterior algebras $\Lambda^*V$ and $\Lambda^*V^*$ of V and its dual $V^*$, and on the symmetric algebra $\S^*V$. The subring of SL(V)-invariants of $\Lambda^*V\otimes\S^*V$ was determined by Dickson and Mui. This paper describes the equivalent, but simpler, calculation of the invariant subring of $\Lambda^*V^*\otimes\S^*V$ as a representation of GL(V)/SL(V).

The main theorem of this paper generalizes a classic Aumann's characterization of compact convex sets, via the acyclicity of their hypersections, to arbitrary weakly closed subsets of a locally convex linear space.

Richard Rado was born in Berlin; he was the second son of Leopold Rado, from Budapest. At one stage of his education he had to decide whether to become a concert pianist or a mathematician. He chose the latter in the belief that he could continue with music as a hobby, but that he could never treat mathematics in that way. He studied at the University of Berlin, but also spent some time in Göttingen. He took a DPh at Berlin with his thesis ‘Studien zur Kombinatorik’ [3] under Issai Schur in 1933. During this period he was also influenced by Erhard Schmidt.

On 16 March 1933, he married Luise Zadek, the elder daughter of Hermann Zadek, whom he had earlier come to know when he needed a partner to play piano duets. It was indeed a remarkable partnership.

As Hitler came to power in 1933, the Rados, being Jewish, made their way to England, Richard having obtained a scholarship of £300 p.a. from Sir Robert Mond through the recommendation of Professor Lindemann (later Lord Cherwell), who had interviewed him in Berlin, to enable him to study at Cambridge.

There is a Weil cohomology theory with coefficients in a pseudofinite field of characteristic zero.

Using basic harmonic analysis on the n-torus, it is shown that the non-commutative n-tori are the only primitive C*-algebras (up to isomorphism) on which the n-torus [ ]n acts freely and ergodically. An explicit construction recovers the generating unitaries in a natural way from the action.