INTERNATIONAL JOURNAL OF GROUP THEORY
Year:2012 | Volume:1 | Issue:2|Papers Count:6

We describe a new type of presentation that, when consistent, describes a polycyclic group. This presentation is obtained by refining a series of normal subgroups with abelian sections.These presentations can be described effectively in computer-algebra-systems like Gap or Magma. We study these presentations and, in particular, we obtain consistency criteria for them. The consistency implementation demonstrates that there are situations where the new method is faster than the existing methods for polycyclic groups.

Given an integer n, we denote by Bn and en the classes of all groups G for which the map fn: g®gn is a monomorphism and an epimorphism of G, respectively. In this paper we give a characterization for groups in Bn and for groups in en. We also obtain an arithmetic description of the set of all integersn such that a group G is in Bn \ en.

Many results were proved on the structure of nite groups with some restrictions on their real elements and on their conjugacy classes. We generalize a few of these to some classes of infinite groups. We study groups in which real elements are central, groups in which real elements are 2-elements, groups in which all non-trivial classes have the same finite size and FC-groups with two non-trivial conjugacy class sizes.

The aim of this paper is to study the (a; g) -prolongation of central extensions. We obtain the obstruction theory for (a; g) -prolongations and classify (a; g) -prolongations thanks to low-dimensional cohomology groups of groups.

Let G be a finite p-soluble group, and P a Sylow p-subgroup of G. It is proved that if all elements of P of order p (or of order £ 4 for p=2) are contained in the k-th term of the upper central series of P, then the p-length of G is at most 2m+1, where m is the greatest integer such that pm-pm-1£k, and the exponent of the image of P in G=Op; p (G) is at most pm. It is also proved that if P is a powerful p-group, then the p-length of G is equal to 1.

A p-group G is p -central if Gp£Z (G), and G is p2 -abelian if (xy) p2 = xp2 yp2 for all x; yÎG. We prove that for G a finite p2 -abelian p -central p -group, excluding certain cases, the order of G divides the order of Aut (G).