Then H need not be divisible, i.e., we need not have pH D H . We only know H is contained in pG. Moreover, for x in H , there need not exist an infinite sequence y1;y2;y3:::: such that py2 D y1, py3 D y2; ::. We only know that there exist arbitrarily long finite such sequences. There does exist a unique maximal divisible subgroup D of M , and D is a subgroup of H . Moreover, M DD ̊N where N is a...

By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups pres...

We shall prove the following theorem concerning compact connected abelian groups: For any element x in a compact connected abelian group there is an element y such that the closure of the cyclic group generated by y is connected and contains x. This is a generalisation of the well known fact that any compact connected group whose topology admits a basis of at most continuum cardinality is monot...

We modify and generalize the construction by McFarland (1973) in two different ways to construct new semiregular divisible difference sets (DDSs) with 21 ~ 0. The parameters of the DDS fall into a family of parameters found in Jungnickel (1982), where his construction is for divisible designs. The final section uses the idea of a K-matrix to find DDSs with a nonelementary abelian forbidden subg...

We give a new construction of difference families generalizing Szekeres’s difference families [17]. As an immediate consequence, we obtain some new examples of difference families with several blocks in multiplicative subgroups of finite fields. We also prove that there exists an infinite family of divisible difference families with two blocks in a unit subgroup of the Galois ring GR(4, n). Fur...

Let A be an abelian variety in algebraically closed field of characteristic 0. We prove that the expansion by a generic divisible subgroup with same torsion exists provided has few algebraic endomorphisms, namely End ( ) = Z . The resulting theory is NSOP1 and not simple. Note there exist varieties any genus.

We give two new constructions of symmetric group divisible designs using divisible difference sets. These constructions generalize the McFarland and Spence constructions of symmetric designs. Divisible designs are of interest in several parts of mathematics: The design theorist needs classes of divisible designs as "ingredients" for recursive constructions of designs. Statisticians use divisibl...

We explore classical (relative) difference sets intersected with the cosets of a subgroup of small index. The intersection sizes are governed by quadratic Diophantine equations. Developing the intersections in the subgroup yields an interesting class of group divisible designs. From this and the Bose-Shrikhande-Parker construction, we obtain some new sets of mutually orthogonal latin squares. W...