Let Sn denote the symmetric group on [n] = {1, . . . , n}. A family I ⊆ Sn is intersecting if any two elements of I have at least one common entry. It is known that the only intersecting families of maximal size in Sn are the cosets of point stabilizers. We show that, under mild restrictions, analogous results hold for the alternating group and the direct product of symmetric groups.

A quantum algorithm for the Hidden Subgroup Problem over the group Z/pZ o Z/qZ is presented. This algorithm, which for certain parameters of the group qualifies as ‘efficient’, generalizes prior work on related semi-direct product groups. 1998 ACM Subject Classification F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems

We analyse volume-preserving actions of product groups on Riemannian manifolds. To this end, we establish a new superrigidity theorem for ergodic cocycles of product groups ranging in linear groups. There are no a priori assumptions on the acting groups, except a spectral gap assumption on their action. Our main application to manifolds concerns irreducible actions of Kazhdan product groups. We...

If the absolute Galois group GK of a field K is a direct product GK = G1 × G2 then one of the factors is prosolvable and either G1 and G2 have coprime order or K is henselian and the direct product decomposition reflects the ramification structure of GK . So, typically, the direct product of two absolute Galois groups is not an absolute Galois group. In contrast, free (profinite) products of ab...

Non-abelian Polish groups arising as countable products of countable groups can be tame in arbitrarily complicated ways. This contrasts with some results of Solecki who revealed a very different picture in the abelian case. 0. Group trees. The class of all Polish (completely metrizable, separable) groups may be naturally divided into two classes. 0.1. Definition. A Polish group G is tame if whe...