On the length of finite groups and of fixed points
Authors:
E. I. Khukhro and P. Shumyatsky
Journal:
Proc. Amer. Math. Soc. 143 (2015), 37813790
MSC (2000):
Primary 20D45
Published electronically:
March 4, 2015
MathSciNet review:
3359570
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Abstract: The generalized Fitting height of a finite group is the least number such that , where the is the generalized Fitting series: and is the inverse image of . It is proved that if admits a soluble group of automorphisms of coprime order, then is bounded in terms of , where is the fixedpoint subgroup, and the number of prime factors of counting multiplicities. The result follows from the special case when is of prime order, where it is proved that . The nonsoluble length of a finite group is defined as the minimum number of nonsoluble factors in a normal series each of whose factors is either soluble or is a direct product of nonabelian simple groups. It is proved that if is a group of automorphisms of of coprime order, then is bounded in terms of and the number of prime factors of counting multiplicities.
 [1]
P.
Hall and Graham
Higman, On the 𝑝length of 𝑝soluble groups and
reduction theorems for Burnside’s problem, Proc. London Math.
Soc. (3) 6 (1956), 1–42. MR 0072872
(17,344b)
 [2]
B.
Huppert, Endliche Gruppen. I, Die Grundlehren der
Mathematischen Wissenschaften, Band 134, SpringerVerlag, BerlinNew York,
1967 (German). MR 0224703
(37 #302)
 [3]
Bertram
Huppert and Norman
Blackburn, Finite groups. III, Grundlehren der Mathematischen
Wissenschaften [Fundamental Principles of Mathematical Sciences],
vol. 243, SpringerVerlag, BerlinNew York, 1982. MR 662826
(84i:20001b)
 [4]
E. I. Khukhro and P. Shumyatsky, Nonsoluble and nonsoluble length of finite groups, to appear in Israel J. Math.; #arXiv:1310.2434#.
 [5]
E.
I. Khukhro and P.
Shumyatsky, Words and pronilpotent subgroups in profinite
groups, J. Aust. Math. Soc. 97 (2014), no. 3,
343–364. MR
3270773, 10.1017/S1446788714000317
 [6]
Hans
Kurzweil, 𝑝Automorphismen von auflösbaren
𝑝′Gruppen, Math. Z. 120 (1971),
326–354 (German). MR 0284503
(44 #1728)
 [7]
John
Thompson, Finite groups with fixedpointfree automorphisms of
prime order, Proc. Nat. Acad. Sci. U.S.A. 45 (1959),
578–581. MR 0104731
(21 #3484)
 [8]
John
G. Thompson, Automorphisms of solvable groups, J. Algebra
1 (1964), 259–267. MR 0173710
(30 #3920)
 [9]
Alexandre
Turull, Fitting height of groups and of fixed points, J.
Algebra 86 (1984), no. 2, 555–566. MR 732266
(85i:20021), 10.1016/00218693(84)900486
 [10]
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
(95f:20038)
 [11]
John
S. Wilson, On the structure of compact torsion groups,
Monatsh. Math. 96 (1983), no. 1, 57–66. MR 721596
(85a:22007), 10.1007/BF01298934
 [1]
 P. Hall and Graham Higman, On the length of soluble groups and reduction theorems for Burnside's problem, Proc. London Math. Soc. (3) 6 (1956), 142. MR 0072872 (17,344b)
 [2]
 B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, SpringerVerlag, BerlinNew York, 1967 (German). MR 0224703 (37 #302)
 [3]
 Bertram Huppert and Norman Blackburn, Finite groups. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 243, SpringerVerlag, BerlinNew York, 1982. MR 662826 (84i:20001b)
 [4]
 E. I. Khukhro and P. Shumyatsky, Nonsoluble and nonsoluble length of finite groups, to appear in Israel J. Math.; #arXiv:1310.2434#.
 [5]
 E. I. Khukhro and P. Shumyatsky, Words and pronilpotent subgroups in profinite groups, J. Aust. Math. Soc. 97 (2014), no. 3, 343364. MR 3270773, https://doi.org/10.1017/S1446788714000317
 [6]
 Hans Kurzweil, Automorphismen von auflösbaren Gruppen, Math. Z. 120 (1971), 326354 (German). MR 0284503 (44 #1728)
 [7]
 John Thompson, Finite groups with fixedpointfree automorphisms of prime order, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 578581. MR 0104731 (21 #3484)
 [8]
 John G. Thompson, Automorphisms of solvable groups, J. Algebra 1 (1964), 259267. MR 0173710 (30 #3920)
 [9]
 Alexandre Turull, Fitting height of groups and of fixed points, J. Algebra 86 (1984), no. 2, 555566. MR 732266 (85i:20021), https://doi.org/10.1016/00218693(84)900486
 [10]
 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, 325331 (English, with Italian summary). MR 1249108 (95f:20038)
 [11]
 John S. Wilson, On the structure of compact torsion groups, Monatsh. Math. 96 (1983), no. 1, 5766. MR 721596 (85a:22007), https://doi.org/10.1007/BF01298934
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Additional Information
E. I. Khukhro
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia – and – University of Lincoln, Lincoln, United Kingdom.
Email:
khukhro@yahoo.co.uk
P. Shumyatsky
Affiliation:
Department of Mathematics, University of Brasilia, DF 70910900, Brazil
Email:
pavel@unb.br
DOI:
https://doi.org/10.1090/S000299392015125737
Keywords:
Finite groups,
nonsoluble length,
generalized Fitting height,
coprime automorphism group
Received by editor(s):
May 2, 2014
Received by editor(s) in revised form:
May 28, 2014
Published electronically:
March 4, 2015
Additional Notes:
This work was supported by CNPqBrazil. The first author thanks CNPqBrazil and the University of Brasilia for support and hospitality that he enjoyed during his visits to Brasilia.
Communicated by:
Pham Huu Tiep
Article copyright:
© Copyright 2015
American Mathematical Society
