| Title:
|
On groups of automorphisms of nilpotent $p$-groups of finite rank (English) |
| Author:
|
Xu, Tao |
| Author:
|
Liu, Heguo |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
1161-1165 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
automorphism |
| Keyword:
|
nilpotent group |
| Keyword:
|
finite rank |
| MSC:
|
20F18 |
| MSC:
|
20F28 |
| idZBL:
|
07285987 |
| idMR:
|
MR4181804 |
| DOI:
|
10.21136/CMJ.2020.0262-19 |
| . |
| Date available:
|
2020-11-18T09:49:54Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148419 |
| . |
| Reference:
|
[1] Dardano, U., Eick, B., Menth, M.: On groups of automorphisms of residually finite groups.J. Algebra 231 (2000), 561-573. Zbl 0967.20022, MR 1778158, 10.1006/jabr.2000.8334 |
| Reference:
|
[2] Guralnick, R. M.: A note on pairs of matrices with rank one commutator.Linear Multilinear Algebra 8 (1979), 97-99. Zbl 0423.15005, MR 552353, 10.1080/03081087908817305 |
| Reference:
|
[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. Zbl 1124.20022, MR 2305790 |
| Reference:
|
[4] Robinson, D. J. S.: Residual properties of some classes of infinite soluble groups.Proc. Lond. Math. Soc., III. Ser. 18 (1968), 495-520. Zbl 0157.05402, MR 228586, 10.1112/plms/s3-18.3.495 |
| Reference:
|
[5] Robinson, D. J. S.: Finiteness Conditions and Generalized Soluble Groups. Part 2.Ergebnisse der Mathematik und ihrer Grenzgebiete 63, Springer, Berlin (1972). Zbl 0243.20033, MR 332990, 10.1007/978-3-662-11747-7 |
| Reference:
|
[6] Robinson, D. J. S.: A Course in the Theory of Groups.Graduate Texts in Mathematics 80, Springer, New York (1982). Zbl 0483.20001, MR 0648604, 10.1007/978-1-4419-8594-1 |
| Reference:
|
[7] Segal, D.: Polycyclic Groups.Cambridge Tracts in Mathematics 82, Cambridge University Press, Cambridge (1983). Zbl 0516.20001, MR 713786, 10.1017/CBO9780511565953 |
| Reference:
|
[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). Zbl 0261.20038, MR 0335656, 10.1007/978-3-642-87081-1 |
| . |
