In 1990, Alon and Kleitman proposed an argument for the sum-free subset problem: every set of n nonzero elements of a finite Abelian group contains a sum-free subset A of size |A| > 27n. In this note, we show that the argument confused two different randomness. It applies only to the finite Abelian group G = (Z/pZ) where p is a prime. For the general case, the problem remains open.

We say that a finite abelian group does not have the Rédei property if it can be expressed as a direct product of two of its subsets such that both subsets contain the identity element and both subsets span the whole group. It will be shown that only a small fraction of the finite abelian groups can have the Rédei property. For groups of odd order an explicit list of the possible exceptions is ...

We define cusp-decomposable manifolds and prove smooth rigidity within this class of manifolds. These manifolds generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, locally symmetric, negatively curved manifolds with cusps. We prove that the group of outer automorphisms of the fundamental group of such a manifold is an e...

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier– Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index.

In this paper, without using the classification of finite simple groups, we determine the structure of  finite simple groups whose Sylow 3-subgroups are of the order 9. More precisely, we classify finite simple groups whose Sylow 3-subgroups are elementary abelian of order 9.

In the case of a locally compact field K, the additive group (K,+) and the multiplicative group (K×, ·) play key roles in the theory. Each is a locally compact abelian group, hence amenable to the methods of Fourier analysis. Moreover, the additive group is self-Pontrjagin dual, and the multiplicative group K× is a target group for class field theory on K: that is, there is a bijective correspo...

For each pointed abelian group (A, c), there is an associated Galkin quandle G(A, c) which is an algebraic structure defined on Z3 ×A that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number o...

The starting point for our discussion is given by the paper [5], where a formula for the number of rooted chains of subgroups of a finite cyclic group is obtained. This leads in [3] to precise expression of the well-known central Delannoy numbers in an arbitrary dimension and has been simplified in [2]. Some steps in order to determine the number of rooted chains of subgroups of a finite elemen...

A residually finite (profinite) group G is just infinite if every non-trivial (closed) normal subgroup of G is of finite index. This paper considers the problem of determining whether a (closed) subgroup H of a just infinite group is itself just infinite. If G is not virtually abelian, we give a description of the just infinite property for normal subgroups in terms of maximal subgroups. In par...

We study the symmetries of periodic solutions obtained from Hopf bifurcation in systems with finite abelian symmetries. The H mod K Theorem gives necessary and sufficient conditions for the existence of periodic solutions with spatial symmetries K and spatio-temporal symmetries H in systems with finite symmetry group Γ. Our main result, the Abelian Hopf H mod K Theorem, gives necessary and suff...