Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
automorphism; nilpotent group; finite rank
automorphism; nilpotent group; finite rank
Summary:
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}\colon 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 $. \item {(i)} $G$ is finite, then $\Gamma $ is an extension of a $p$-group by an abelian group. \item {(ii)} $G$ is infinite, then $\Gamma $ is soluble and abelian-by-finite.
References:
[1] Dardano, U., Eick, B., Menth, M.: 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] Guralnick, R. M.: 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] Liu, H. G., Zhang, J. P.: On $p$-automorphisms of a nilpotent $p$-group with finite rank. Acta Math. Sin., Chin. Ser. 50 (2007), 11-16 Chinese. MR 2305790 | Zbl 1124.20022
[4] Robinson, D. J. S.: 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] Robinson, D. J. S.: 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] Robinson, D. J. S.: 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] Segal, D.: Polycyclic Groups. Cambridge Tracts in Mathematics 82, Cambridge University Press, Cambridge (1983). DOI 10.1017/CBO9780511565953 | MR 713786 | Zbl 0516.20001
[8] Wehrfritz, B. A. F.: 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
Search
Partner of
