Finite $p$-groups with a Frobenius group of automorphisms whose kernel is a cyclic $p$-group
HTML articles powered by AMS MathViewer
- by E. I. Khukhro and N. Yu. Makarenko PDF
- Proc. Amer. Math. Soc. 143 (2015), 1837-1848 Request permission
Abstract:
Suppose that a finite $p$-group $P$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ that is a cyclic $p$-group and with complement $H$. It is proved that if the fixed-point subgroup $C_P(H)$ of the complement is nilpotent of class $c$, then $P$ has a characteristic subgroup of index bounded in terms of $c$, $|C_P(F)|$, and $|F|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. Examples show that the condition of $F$ being cyclic is essential. The proof is based on a Lie ring method and a theorem of the authors and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of automorphisms $FH$. It is also proved that $P$ has a characteristic subgroup of $(|C_P(F)|, |F|)$-bounded index whose order and rank are bounded in terms of $|H|$ and the order and rank of $C_P(H)$, respectively, and whose exponent is bounded in terms of the exponent of $C_P(H)$.References
- J. L. Alperin, Automorphisms of solvable groups, Proc. Amer. Math. Soc. 13 (1962), 175–180. MR 142639, DOI 10.1090/S0002-9939-1962-0142639-3
- J. D. Dixon, M. P. F. Du Sautoy, A. Mann, D. Segal, Analytic pro-$p$ groups, 2nd Ed., Cambridge Univ. Press, 2003.
- P. Hall, A Contribution to the Theory of Groups of Prime-Power Order, Proc. London Math. Soc. (2) 36 (1934), 29–95. MR 1575964, DOI 10.1112/plms/s2-36.1.29
- 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
- Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR 650245
- A. Jaikin-Zapirain, On almost regular automorphisms of finite $p$-groups, Adv. Math. 153 (2000), no. 2, 391–402. MR 1770935, DOI 10.1006/aima.1999.1911
- V. A. Kreknin, Solvability of Lie algebras with a regular automorphism of finite period, Dokl. Akad. Nauk SSSR 150 (1963), 467–469 (Russian). MR 0157990
- V. A. Kreknin and A. I. Kostrikin, Lie algebras with regular automorphisms, Dokl. Akad. Nauk SSSR 149 (1963), 249–251 (Russian). MR 0146230
- E. I. Khukhro, Finite $p$-groups admitting an automorphism of order $p$ with a small number of fixed points, Mat. Zametki 38 (1985), no. 5, 652–657, 795 (Russian). MR 819622
- E. I. Khukhro, Finite $p$-groups that admit $p$-automorphisms with a small number of fixed points, Mat. Sb. 184 (1993), no. 12, 53–64 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 80 (1995), no. 2, 435–444. MR 1254804, DOI 10.1070/SM1995v080n02ABEH003532
- Evgenii I. Khukhro, Nilpotent groups and their automorphisms, De Gruyter Expositions in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1993. MR 1224233, DOI 10.1515/9783110846218
- 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, The rank and order of a finite group that admits a Frobenius group of automorphisms, Algebra Logika 52 (2013), no. 1, 99–108, 124, 127 (Russian, with English and Russian summaries); English transl., Algebra Logic 52 (2013), no. 1, 72–78. MR 3113801, DOI 10.1007/s10469-013-9221-1
- E. I. Khukhro and N. Yu. Makarenko, Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms, J. Algebra 386 (2013), 77–104. MR 3049577, DOI 10.1016/j.jalgebra.2013.04.008
- Evgeny Khukhro, Natalia Makarenko, and Pavel Shumyatsky, Frobenius groups of automorphisms and their fixed points, Forum Math. 26 (2014), no. 1, 73–112. MR 3176625, DOI 10.1515/form.2011.152
- Ian Kiming, Structure and derived length of finite $p$-groups possessing an automorphism of $p$-power order having exactly $p$ fixpoints, Math. Scand. 62 (1988), no. 2, 153–172. MR 964222, DOI 10.7146/math.scand.a-12212
- Hans Kurzweil, $p$-Automorphismen von auflösbaren $p^{\prime }$-Gruppen, Math. Z. 120 (1971), 326–354 (German). MR 284503, DOI 10.1007/BF01109999
- N. Yu. Makarenko, Almost regular automorphisms of prime order, Sibirsk. Mat. Zh. 33 (1992), no. 5, 206–208, 224 (Russian, with Russian summary); English transl., Siberian Math. J. 33 (1992), no. 5, 932–934 (1993). MR 1197087, DOI 10.1007/BF00971002
- N. Yu. Makarenko and E. I. Khukhro, Lie algebras admitting a metacyclic Frobenius group of automorphisms, Sibirsk. Mat. Zh. 54, no. 1 (2013), 131–149; English transl., Siberian Math. J. 54 (2013), 50–64.
- N. Yu. Makarenko, E. I. Khukhro, and P. Shumyatskiĭ, Fixed points of Frobenius groups of automorphisms, Dokl. Akad. Nauk 437 (2011), no. 1, 20–23 (Russian); English transl., Dokl. Math. 83 (2011), no. 2, 152–154. MR 2849320, DOI 10.1134/S1064562411020050
- Alexander Lubotzky and Avinoam Mann, Powerful $p$-groups. I. Finite groups, J. Algebra 105 (1987), no. 2, 484–505. MR 873681, DOI 10.1016/0021-8693(87)90211-0
- Susan McKay, On the structure of a special class of $p$-groups, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 152, 489–502. MR 916230, DOI 10.1093/qmath/38.4.489
- Yuri Medvedev, $p$-groups, Lie $p$-rings and $p$-automorphisms, J. London Math. Soc. (2) 58 (1998), no. 1, 27–37. MR 1666066, DOI 10.1112/S0024610798006449
- Yuri Medvedev, $p$-divided Lie rings and $p$-groups, J. London Math. Soc. (2) 59 (1999), no. 3, 787–798. MR 1709080, DOI 10.1112/S0024610799007310
- Aner Shalev, On almost fixed point free automorphisms, J. Algebra 157 (1993), no. 1, 271–282. MR 1219668, DOI 10.1006/jabr.1993.1100
- A. Shalev and E. I. Zel′manov, Pro-$p$ groups of finite coclass, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 417–421. MR 1151320, DOI 10.1017/S0305004100075514
- John G. Thompson, Automorphisms of solvable groups, J. Algebra 1 (1964), 259–267. MR 173710, DOI 10.1016/0021-8693(64)90022-5
- Alexandre Turull, Fitting height of groups and of fixed points, J. Algebra 86 (1984), no. 2, 555–566. MR 732266, DOI 10.1016/0021-8693(84)90048-6
Additional Information
- E. I. Khukhro
- Affiliation: Sobolev Institute of Mathematics, Novosibirsk, 630 090, Russia
- MR Author ID: 227765
- Email: khukhro@yahoo.co.uk
- N. Yu. Makarenko
- Affiliation: Sobolev Institute of Mathematics, Novosibirsk, 630 090, Russia
- Address at time of publication: Université de Haute Alsace, Mulhouse, 68093, France
- Email: natalia_makarenko@yahoo.fr
- Received by editor(s): February 14, 2013
- Received by editor(s) in revised form: May 29, 2013
- Published electronically: January 22, 2015
- Additional Notes: The first author was supported by the Russian Science Foundation, project no. 14-21-00065
The second author was supported in part by the Russian Foundation for Basic Research, project no. 13-01-00505 - Communicated by: Pham Huu Tiep
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 1837-1848
- MSC (2010): Primary 20D45; Secondary 17B40, 17B70, 20D15
- DOI: https://doi.org/10.1090/S0002-9939-2015-12287-3
- MathSciNet review: 3314095
Dedicated: Dedicated to Victor Danilovich Mazurov on the occasion of his 70th birthday