Profinite groups with an automorphism whose fixed points are right Engel
HTML articles powered by AMS MathViewer
- by C. Acciarri, E. I. Khukhro and P. Shumyatsky PDF
- Proc. Amer. Math. Soc. 147 (2019), 3691-3703 Request permission
Abstract:
An element $g$ of a group $G$ is said to be right Engel if for every $x\in G$ there is a number $n=n(g,x)$ such that $[g,{}_{n}x]=1$. We prove that if a profinite group $G$ admits a coprime automorphism $\varphi$ of prime order such that every fixed point of $\varphi$ is a right Engel element, then $G$ is locally nilpotent.References
- Reinhold Baer, Engelsche elemente Noetherscher Gruppen, Math. Ann. 133 (1957), 256–270 (German). MR 86815, DOI 10.1007/BF02547953
- Y. A. Bahturin and M. V. Zaicev, Identities of graded algebras, J. Algebra 205 (1998), no. 1, 1–12. MR 1631298, DOI 10.1006/jabr.1997.7017
- R. G. Burns and Yuri Medvedev, A note on Engel groups and local nilpotence, J. Austral. Math. Soc. Ser. A 64 (1998), no. 1, 92–100. MR 1490149, DOI 10.1017/S1446788700001324
- K. W. Gruenberg, The Engel structure of linear groups, J. Algebra 3 (1966), 291–303. MR 195956, DOI 10.1016/0021-8693(66)90003-2
- P. Hall, Some sufficient conditions for a group to be nilpotent, Illinois J. Math. 2 (1958), 787–801. MR 105441, DOI 10.1215/ijm/1255448649
- Hermann Heineken, Eine Bemerkung über engelsche Elemente, Arch. Math. (Basel) 11 (1960), 321 (German). MR 125153, DOI 10.1007/BF01236951
- Graham Higman, Groups and rings having automorphisms without non-trivial fixed elements, J. London Math. Soc. 32 (1957), 321–334. MR 89204, DOI 10.1112/jlms/s1-32.3.321
- John L. Kelley, General topology, Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]. MR 0370454
- E. I. Khukhro, $p$-automorphisms of finite $p$-groups, London Mathematical Society Lecture Note Series, vol. 246, Cambridge University Press, Cambridge, 1998. MR 1615819, DOI 10.1017/CBO9780511526008
- E. I. Khukhro and P. Shumyatsky, Bounding the exponent of a finite group with automorphisms, J. Algebra 212 (1999), no. 1, 363–374. MR 1670607, DOI 10.1006/jabr.1998.7609
- Michel Lazard, Groupes analytiques $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603 (French). MR 209286
- V. Linchenko, Identities of Lie algebras with actions of Hopf algebras, Comm. Algebra 25 (1997), no. 10, 3179–3187. MR 1465109, DOI 10.1080/00927879708826047
- Alexander Lubotzky and Avinoam Mann, Powerful $p$-groups. II. $p$-adic analytic groups, J. Algebra 105 (1987), no. 2, 506–515. MR 873682, DOI 10.1016/0021-8693(87)90212-2
- Derek J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR 1357169, DOI 10.1007/978-1-4419-8594-1
- Aner Shalev, Polynomial identities in graded group rings, restricted Lie algebras and $p$-adic analytic groups, Trans. Amer. Math. Soc. 337 (1993), no. 1, 451–462. MR 1093218, DOI 10.1090/S0002-9947-1993-1093218-X
- Pavel Shumyatsky, Applications of Lie ring methods to group theory, Nonassociative algebra and its applications (São Paulo, 1998) Lecture Notes in Pure and Appl. Math., vol. 211, Dekker, New York, 2000, pp. 373–395. MR 1755367
- Pavel Shumyatsky and Danilo Sanção da Silveira, On finite groups with automorphisms whose fixed points are Engel, Arch. Math. (Basel) 106 (2016), no. 3, 209–218. MR 3463708, DOI 10.1007/s00013-015-0866-y
- John Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 578–581. MR 104731, DOI 10.1073/pnas.45.4.578
- John S. Wilson, Profinite groups, London Mathematical Society Monographs. New Series, vol. 19, The Clarendon Press, Oxford University Press, New York, 1998. MR 1691054
- John S. Wilson and Efim I. Zelmanov, Identities for Lie algebras of pro-$p$ groups, J. Pure Appl. Algebra 81 (1992), no. 1, 103–109. MR 1173827, DOI 10.1016/0022-4049(92)90138-6
- Efim Zelmanov, Nil rings and periodic groups, KMS Lecture Notes in Mathematics, Korean Mathematical Society, Seoul, 1992. With a preface by Jongsik Kim. MR 1199575
- E. I. Zel′manov, Lie ring methods in the theory of nilpotent groups, Groups ’93 Galway/St. Andrews, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 212, Cambridge Univ. Press, Cambridge, 1995, pp. 567–585. MR 1337299, DOI 10.1017/CBO9780511629297.023
- Efim Zelmanov, Lie algebras and torsion groups with identity, J. Comb. Algebra 1 (2017), no. 3, 289–340. MR 3681577, DOI 10.4171/JCA/1-3-2
Additional Information
- C. Acciarri
- Affiliation: Department of Mathematics, University of Brasilia, Brasilia DF 70910-900, Brazil
- MR Author ID: 933258
- Email: C.Acciarri@mat.unb.br
- E. I. Khukhro
- Affiliation: Charlotte Scott Research Centre for Algebra, University of Lincoln, Lincoln, LN6 7TS, United Kingdom; and Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
- MR Author ID: 227765
- Email: khukhro@yahoo.co.uk
- P. Shumyatsky
- Affiliation: Department of Mathematics, University of Brasilia, Brasilia DF 70910-900, Brazil
- MR Author ID: 250501
- Email: p.shumyatsky@mat.unb.br
- Received by editor(s): August 6, 2018
- Received by editor(s) in revised form: December 10, 2018
- Published electronically: April 8, 2019
- Additional Notes: The first and third authors were supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.
The second author was supported by the Russian Science Foundation, project no. 14-21-00065. The second author thanks CNPq-Brazil and the University of Brasilia for support and hospitality that he enjoyed during his visit to Brasilia. - Communicated by: Pham Huu Tiep
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3691-3703
- MSC (2010): Primary 20E18, 20E36; Secondary 20F45, 20F40, 20D15, 20F19
- DOI: https://doi.org/10.1090/proc/14519
- MathSciNet review: 3993763