It is a well known theorem that an abelian group G satisfying G = nG for every positive integer n is a direct summand of every abelian group H which contains G as a subgroup. It is the object of this note to generalize this theorem to abelian groups admitting a ring of operators, and to show that the corresponding conditions are not only sufficient but are at the same time necessary. Finally we...

We give an elementary proof of the following result: If G is a compact non-zero Abelian group with dual isomorphic to a subgroup of Q, such that U u (U) = G \ G(2) and U n (U) = 0 for some open subset U c G, where G(2) = {a E G: 2a = O}, then G is topologically isomorphic with T.

We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, have positive first Betti number, a homomorphisms onto non-Abelian free group. For we determine minimal neat subgroup. Finally, answer question raised by Stover describe an infinite tower ball quotients all with single cusp.

Duality arguments suggest the existence of massless magnetic monopoles in gauge theories with the symmetry broken to a non-Abelian subgroup. I discuss how these arise and show how they are manifested as clouds of massless fields surrounding massive monopoles. The dynamics of these clouds is discussed, and the scattering of massless monopole clouds and massive monopoles is described.

Let G be a finite group with a non-abelian minimal normal subgroup N which is a direct product of the simple group X. The maximal subgroups of G which complement N and their conjugacy classes are parametrised in terms of certain homomorphisms taking values in AutX and satisfying particular conditions.

Now that the classification of finite simple groups is complete, it is logical to look at the extension problem. An important special case to consider is when M is a minimal normal subgroup of G and both G/M and M are known groups. If M is abelian, various techniques have been used to derive information about G. Indeed, almost the entire theory of finite solvable groups can be said to rest upon...