International Journal of Group Theory
Volume:1 Issue: 2, Jun 2012

We describe a new type of presentation that، when consistent، describes a polycyclic group. This presentation is obtained by re ning a series of normal subgroups with abelian sections. These presentations can be described e ectively in computeralgebra- 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.

Let n be an integer. A group G is said to be n-abelian if the map phi_n that sends g to g^n is an endomorphism of G. Then (xy) ^n=x^ny^n for all x،y in G، from which it follows [x^n،y] = [x،y] ^n= [x،y^n]. It is also easy to see that a group G is n-abelian if and only if it is (1-n) -abelian. If nneq 0،1 and G is an n-abelian group، then the quotient group G/Z (G) has finite exponent dividing n (n-1). This implies that every torsion-free n-abelian group is abelian. We denote by B_n and C_n the classes of all groups G for which phi_n is a monomorphism and an epimorphism of G، respectively. Then B_0=C_0 contains only the trivial group، B_1=C_1 is the class of all groups، and B_-1=C_-1 is the class of all abelian groups. Furthermore، with |n|>1، G is in B_n if and only if G is an n-abelian group having no elements of order dividing |n|. Similarly، G is in C_n if and only if G is n-abelian and for every g in G there exists an element x in G such that g=x^n. We also set A_n=B_ncap C_n. In this paper we give a characterization for groups in B_n and for groups in C_n. We also obtain an arithmetic description of the set of all integers n such that a group G is in A_n.

Many results were proved on the structure of finite 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 $ (alpha، gamma) $-prolongation of central extensions. We obtain the obstruction theory for $ (alpha، gamma) $-prolongations and classify $ (alpha، gamma) $-prolongations thanks to low-dimensional cohomology groups of groups.

Let $G$ be a finite $p$-soluble group، and $P$ a Sylow $p$-sub-group of $G$. It is proved that if all elements of $P$ of order $p$ (or of order ${}leq 4$ 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 $p^m-p^ {m-1} leq k$، and the exponent of the image of $P$ in $G/O_ {p''،p} (G) $ is at most $p^m$. 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 is p-central if the central quotient has exponent p، and G is (p^2) -abelian if (xy) ^ {p^{2}}= (x^ {p^2}) (y^ {p^2}) for all x،y in G. We prove that for G a finite (p^2) -abelian p-central p-group، excluding certain cases، the order of G divides the order of Aut (G).