Narrow Results

A polycyclic group $G$ is called an $NAF{Q}_n$-group if every normal abelian subgroup of any finite quotient of $G$ is generated by $n$, or fewer, elements and $n$ is the least integer with this property. In this paper, the structure of $NAF{Q}_1$-groups and $NAF{Q}_2$-groups is determined.

A polycyclic group $G$ is called a $NAF{Q}_n$-group ($AF{Q}_n$-group) if every normal abelian subgroup (abelian subgroup) of any finite quotient of $G$ is generated by $n$, or fewer, elements and $n$ is the least integer with this property. In this paper, we describe the structures of $NAF{Q}_3$-groups and $AF{Q}_3$-groups, and bound the number of generators of $AF{Q}_3$-groups and the derived lengths of $NAF{Q}_3$-groups, which is a continuation of [H. G. Liu, F. Zhou and T. Xu, On some polycyclic groups with small Hirsch length, J. Algebra Appl. 16(11) (2017) 17502371–175023710].

In this paper we show that some finiteness properties on a centralizer of a particular subgroup can be inherited by the whole group. Among other things, we prove the following characterization of polycyclic groups: a soluble group G is polycyclic if and only if it contains a finitely generated subgroup H, formed by bounded left Engel elements, whose centralizer C_G(H) is polycyclic. In the context of Černikov groups we obtain a more general result: a radical group is a Černikov group if and only if it contains a finitely generated subgroup, formed by left Engel elements, whose centralizer is a Černikov group. The aforementioned results generalize a theorem by Onishchuk and Zaĭtsev about the centralizer of a finitely generated subgroup in a nilpotent group.

The structure of groups in which many subgroups have a certain property χ has been investigated for several choices of the property χ. In particular, groups whose non-normal subgroups are supersoluble are studied in this paper. Moreover, groups with only finitely many normalizers of non-supersoluble groups are considered.

Let $A$ be a completely decomposable homogeneous torsion-free abelian group of rank $n$ ($n\ge 2$). Let ${\mathrm{\Theta }}_m(n)=A⋊⟨\alpha ⟩$ be the split extension of $A$ by an automorphism $\alpha$ which is a cyclic permutation of the direct components twisted by a rational integer $m$. Then ${\mathrm{\Theta }}_m(n)$ is an infinite soluble group. In this paper, the residual finiteness of ${\mathrm{\Theta }}_m(n)$ is investigated.