Near-rings of continuous functions on compact abelian groups

In this paper we show that, in the near-ring N(G) of all continuous selfmaps of a compact abelian groupG with nontrivial connected components, the intersection of all nonzero ideals consists of all functions which are homotopic to the constant endomorphism ofG.     

Linking systems in nonelementary abelian groups

Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to important applications. In this paper, we define linking systems, collections of difference sets that correspond to systems of linked designs, and we construct linking systems in a variety of nonelementary abelian groups using Galois rings, partial difference sets, and a product construction. We include some partial results in the final section.     

A simple proof of the bernoullicity of ergodic automorphisms on compact abelian groups

Recently it was proved by D. Lind, and G. Miles and K. Thomas that every ergodic automorphism of a compact metric abelian group is Bernoullian. They reduce the problem to the finite-dimensional compact connected abelian group (solenoidal group), and then they use difficult methods in proving the case. By using ideas of Y. Katznelson we can give a proof, which is much simpler than the other extant proofs, for the solenoidal case.     

Projective free algebras of continuous functions on compact abelian groups

It is proved that the Wiener algebra of functions on a connected compact abelian group whose Bohr-Fourier spectra are contained in a fixed subsemigroup of the (additive) dual group, is projective free. The semigroup is assumed to contain zero and have the property that it does not contain both a nonzero element and its opposite. The projective free property is proved also for the algebra of continuous functions with the same condition on their Bohr-Fourier spectra. As an application, the connected components of the set of factorable matrices are described. The proofs are based on a key result on homotopies of continuous maps on the maximal ideal spaces of the algebras under consideration.     

Infinitary model theory of abelian groups

The paper is a survey of results in the model theory of abelian groups, dealing with two sorts of problems: finding invariants which classify groups up toL _λκ-equivalence; and determining whether certain classes of groups are definable inL _λκ. Research supported by NSF grant GP 43910     

Random systems of equations in free abelian groups

We study the solvability of random systems of equations on the free abelian group ? ^m of rank m. Denote by SAT(? ^m , k, n) and \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) the sets of all systems of n equations of k unknowns in ? ^m satisfiable in ? ^m and ? ^m respectively. We prove that the asymptotic density \(\rho \left( {SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)} \right)\) of the set \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) equals 1 for n ≤ k and 0 for n > k. As regards, SAT(? ^m , k, n) for n < k, some new estimates are obtained for the lower and upper asymptotic densities and it is proved that they lie between (Π _j=k?n+1 ^k ζ(j))^?1 and \(\left( {\tfrac{{\zeta (k + m)}} {{\zeta (k)}}} \right)^n\) , where ξ(s) is the Riemann zeta function. For n ≤ k, a connection is established between the asymptotic density of SAT(? ^m , k, n) and the sums of inverse greater divisors over matrices of full rank. Starting from this result, we make a conjecture about the asymptotic density of SAT(? ^m , n, n). We prove that ρ(SAT(? ^m , k, n)) = 0 for n > k.     

A structure theorem for abelian quasi-ordered groups

We introduce a notion of compatible quasi-ordered groups which unifies valued and ordered abelian groups. It was proved by S.M. Fakhruddin that a compatible quasi-order on a field is always either an order or a valuation. We show here that the group case is more complicated than the field case and describe the general structure of a compatible quasi-ordered abelian group. We then define a notion of Hahn product of compatible quasi-ordered groups and generalize Hahn's embedding theorem to quasi-ordered groups. We also develop a notion of quasi-order-minimality and establish a connection with C-minimality, thus answering a question of F. Delon. Finally, we use compatible quasi-ordered groups to give an example of a C-minimal group which is neither an ordered nor a valued group.     

2-blocks with abelian defect groups

We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan?s conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan matrices are bounded in terms of the defect for arbitrary abelian 2-groups. We also show that a block with defect groups of the form _C2m×_C2m$C_{2^m}\times C_{2^m}$ for m?2$m?2$ has one of two Morita equivalence types and hence is Morita equivalent to the Brauer correspondent block of the normaliser of a defect group. This completes the analysis of the Morita equivalence types of 2-blocks with abelian defect groups of rank 2, from which we conclude that Donovan?s conjecture holds for such 2-groups. A further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16. The proof uses the classification of finite simple groups.     

The p-local ranks of finite simple groups with abelian Sylow p-subgroups

As a continuation of previous work, in this paper, we mainly study the finite simple groups which have abelian Sylow subgroups in term of p-local rank, especially a group theoretic characterization will be given.