Czechoslovak Mathematical Journal, Vol. 70, No. 4, pp. 1161-1165, 2020


On groups of automorphisms of nilpotent $p$-groups of finite rank

Tao Xu, Heguo Liu

Received June 8, 2019.   Published online April 22, 2020.

Abstract:  Let $\alpha$ and $\beta$ be automorphisms of a nilpotent $p$-group $G$ of finite rank. Suppose that $\langle(\alpha\beta(g))(\beta\alpha(g))^{-1} g\in G\rangle$ is a finite cyclic subgroup of $G$, then, exclusively, one of the following statements holds for $G$ and $\Gamma$, where $\Gamma$ is the group generated by $\alpha$ and $\beta$. (i) $G$ is finite, then $\Gamma$ is an extension of a $p$-group by an abelian group. (ii) $G$ is infinite, then $\Gamma$ is soluble and abelian-by-finite.
Keywords:  automorphism; nilpotent group; finite rank
Classification MSC:  20F18, 20F28


References:
[1] U. Dardano, B. Eick, M. Menth: On groups of automorphisms of residually finite groups. J. Algebra 231 (2000), 561-573. DOI 10.1006/jabr.2000.8334 | MR 1778158 | Zbl 0967.20022
[2] R. M. Guralnick: A note on pairs of matrices with rank one commutator. Linear Multilinear Algebra 8 (1979), 97-99. DOI 10.1080/03081087908817305 | MR 552353 | Zbl 0423.15005
[3] H. G. Liu, J. P. Zhang: On $p$-automorphisms of a nilpotent $p$-group with finite rank. Acta Math. Sin., Chin. Ser. 50 (2007), 11-16. (In Chinese.) MR 2305790 | Zbl 1124.20022
[4] D. J. S. Robinson: Residual properties of some classes of infinite soluble groups. Proc. Lond. Math. Soc., III. Ser. 18 (1968), 495-520. DOI 10.1112/plms/s3-18.3.495 | MR 228586 | Zbl 0157.05402
[5] D. J. S. Robinson: Finiteness Conditions and Generalized Soluble Groups. Part 2. Ergebnisse der Mathematik und ihrer Grenzgebiete 63, Springer, Berlin (1972). DOI 10.1007/978-3-662-11747-7 | MR 332990 | Zbl 0243.20033
[6] D. J. S. Robinson: A Course in the Theory of Groups. Graduate Texts in Mathematics 80, Springer, New York (1982). DOI 10.1007/978-1-4419-8594-1 | MR 0648604 | Zbl 0483.20001
[7] D. Segal: Polycyclic Groups. Cambridge Tracts in Mathematics 82, Cambridge University Press, Cambridge (1983). DOI 10.1017/CBO9780511565953 | MR 713786 | Zbl 0516.20001
[8] B. A. F. Wehrfritz: Infinite Linear Groups. An Account of the Group-Theoretic Properties of Infinite Groups of Matrices. Ergebnisse der Mathematik und ihrer Grenzgebiete 76, Springer, Berlin (1973). DOI 10.1007/978-3-642-87081-1 | MR 0335656 | Zbl 0261.20038

Affiliations:   Tao Xu, Department of Science, Hebei University of Engineering, No. 19 Taiji Road, Handan, 056038, Hebei, P. R. China, e-mail: gtxutao@163.com; Heguo Liu, Department of Mathematics, Hubei University, No. 368 Youyi Road, Wuhan, 430062, Hubei, P. R. China, e-mail: ghliu@hubu.edu.cn


 
PDF available at: