Volume 16, Issue 2

Abstract. We show that the blockwise version of the Alperin weight conjecture is true if for every finite non-abelian simple group a set of conditions holds. Furthermore we prove that several series of simple groups satisfy these assumptions. This refines recent work of Navarro and Tiep, who proved an analogous reduction theorem for the non-blockwise version of the Alperin weight conjecture.

Abstract. A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably-generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tuples depend only on the Ulm invariants of the module in question (and not on the underlying ring). A complete description of the poset of orbits of elements in terms of the Ulm invariants of the module is given. The relationship between this description of orbits and a very different-looking one obtained by Dutta and Prasad for torsion modules of bounded order is explained.

Abstract. If G is a finite group, then denotes the fraction of ordered pairs of elements of G which commute. We show that if is a limit point of the function Pr on finite groups, then and there exists an such that for any finite group  G . These results lend support to some old conjectures of Keith Joseph.

Abstract. A subgroup H of a group G is called a TI-subgroup if , for all , and a group is called a CTI-group if all of its cyclic subgroups are TI-subgroups. In this paper, we determine the structure of non-nilpotent CTI-groups. Also we will show that if G is a nilpotent CTI-group, then G is either a Hamiltonian group or a non-abelian p -group.

Abstract. A normal subgroup K of a finite group G is said to be hypercyclically embedded in G if every chief factor of G below K is cyclic. A subgroup H has the cover-avoidance property in G if H either covers or avoids every chief factor of G . In this paper we connect these two concepts and give a new characterization of normal hypercyclically embedded subgroups. Our main result is that a normal subgroup K is hypercyclically embedded in G if and only if the members of a certain class of subgroups of K have the cover-avoidance property in G .

Abstract. A characterization is given of finite groups of order indivisible by the fourth power of any prime, which shows that such a group has the following form: or , where and D are all nilpotent, M is perfect, and E is abelian. Further, the semi-direct products involved in these groups are characterized.

Abstract. It is proved that every group of the form with subgroups A and B , each of which is either abelian or generalized dihedral, is soluble.