Research ArticleNo Access

On some polycyclic groups with small Hirsch length II

Heguo Liu

Department of Mathematics, Hubei University, Wuhan 430062, P. R. China

E-mail Address: ghliu@hubu.edu.cn

Search for more papers by this author

Fang Zhou

Department of Mathematics, Taiyuan Normal University, Taiyuan 030012, P. R. China

E-mail Address: fangzhou20@aliyun.com

Search for more papers by this author

, and

Tao Xu

Department of Science, Hebei University of Engineering, Handan 056038, P. R. China

E-mail Address: gtxutao@163.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S021949881950169XCited by:0 (Source: Crossref)

Abstract

A polycyclic group $G$ $G$ is called a $N A F Q_{n}$ -group ( $A F Q_{n}$ -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 $N A F Q_{3}$ -groups and $A F Q_{3}$ -groups, and bound the number of generators of $A F Q_{3}$ -groups and the derived lengths of $N A F Q_{3}$ -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].

Communicated by Shun-Jen Cheng

Keywords:

AMSC: 20F19

References

1. N. Blackburn , Nilpotent groups in which the derived group has two generators, J. London Math. Soc. 35 (1960) 33–35. Google Scholar
2. Y. Fan , On the normal series of $D_{2}$ -groups, Chin. Ann. Math. 8A(5) (1987) 626–631. Google Scholar
3. L. G. Kovacs , On finite soluble groups, Math. Z. 103 (1968) 37–39. Web of Science, Google Scholar
4. P. A. Linnell and D. Warhurst , Bounding the number of generators of a polycyclic group, Arch. Math. 37 (1981) 7–17. Web of Science, Google Scholar
5. H. G. Liu, F. Zhou and T. Xu , On some polycyclic groups with small Hirsch length, J. Algebra Appl. 16(11) (2017) 17502371–175023710. Google Scholar
6. J. J. McCutcheon , On certain polycyclic groups, Bull. London Math. Soc. 1 (1969) 179–186. Google Scholar
7. D. J. S. Robinson , A Course in the Theory of Group (Springer, New York, 1996). Google Scholar
8. K. Tahara , On the finite subgroups of GL(3,Z), Nagoya Math. J. 41(1971) 169–209. Web of Science, Google Scholar