Finite groups and their coprime automorphisms
Authors:
Emerson de Melo and Pavel Shumyatsky
Journal:
Proc. Amer. Math. Soc. 145 (2017), 3755-3760
MSC (2010):
Primary 20D45
DOI:
https://doi.org/10.1090/proc/13550
Published electronically:
March 27, 2017
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let 
be a prime and 
a finite group of exponent 
acting by automorphisms on a finite 
-group 
. Assume that 
has order at least 
and 
is nilpotent of class at most 
for any 
. It is shown that 
is nilpotent with class bounded solely in terms of 
and 
.
- [1] Cristina Acciarri and Pavel Shumyatsky, Double automorphisms of graded Lie algebras, J. Algebra 387 (2013), 1-10. MR 3056683, https://doi.org/10.1016/j.jalgebra.2012.12.023
- [2] Chong-yun Chao, Some characterizations of nilpotent Lie algebras, Math. Z. 103 (1968), 40-42. MR 0223415
- [3]
Emerson de Melo, On
-groups of maximal class as groups of automorphisms, J. Algebra 440 (2015), 100-112. MR 3373389, https://doi.org/10.1016/j.jalgebra.2015.06.002
- [4] Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
- [5] Robert Guralnick and Pavel Shumyatsky, Derived subgroups of fixed points, Israel J. Math. 126 (2001), 345-362. MR 1882044, https://doi.org/10.1007/BF02784161
- [6] P. Hall, Some sufficient conditions for a group to be nilpotent, Illinois J. Math. 2 (1958), 787-801. MR 0105441
- [7] Evgenii I. Khukhro, Nilpotent groups and their automorphisms, de Gruyter Expositions in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1993. MR 1224233
- [8] E. I. Khukhro and P. V. Shumyatskiĭ, On fixed points of automorphisms of Lie rings and of locally finite groups, Algebra i Logika 34 (1995), no. 6, 706-723, 730 (Russian, with Russian summary); English transl., Algebra and Logic 34 (1995), no. 6, 395-405 (1996). MR 1400710, https://doi.org/10.1007/BF00739335
- [9] 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
- [10] Pavel Shumyatsky, On periodic solvable groups having automorphisms with nilpotent fixed point groups, Israel J. Math. 87 (1994), no. 1-3, 111-116. MR 1286819, https://doi.org/10.1007/BF02772987
- [11] Pavel Shumyatsky, Finite groups and the fixed points of coprime automorphisms, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3479-3484 (electronic). MR 1860479, https://doi.org/10.1090/S0002-9939-01-06125-1
- [12] John Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 578-581. MR 0104731
- [13] Yan Ming Wang and Zhong Mu Chen, Solubility of finite groups admitting a coprime order operator group, Boll. Un. Mat. Ital. A (7) 7 (1993), no. 3, 325-331 (English, with Italian summary). MR 1249108
- [14] J. N. Ward, On finite groups admitting automorphisms with nilpotent fixed-point group, Bull. Austral. Math. Soc. 5 (1971), 281-282. MR 0296163
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Additional Information
Emerson de Melo
Affiliation:
Department of Mathematics, University of Brasília, Brasília-DF, 70910-900, Brazil
Email:
emerson@mat.unb.br
Pavel Shumyatsky
Affiliation:
Department of Mathematics, University of Brasília, Brasília-DF, 70910-900, Brazil
Email:
pavel@unb.br
DOI:
https://doi.org/10.1090/proc/13550
Keywords:
$p$-groups,
automorphisms,
fixed-point
Received by editor(s):
June 3, 2016
Received by editor(s) in revised form:
September 19, 2016, October 7, 2016, and October 12, 2016
Published electronically:
March 27, 2017
Additional Notes:
This research was supported by FAPDF and CNPq-Brazil
Communicated by:
Pham Huu Tiep
Article copyright:
© Copyright 2017
American Mathematical Society
