On the length of finite groups and of fixed points
HTML articles powered by AMS MathViewer
- by E. I. Khukhro and P. Shumyatsky PDF
- Proc. Amer. Math. Soc. 143 (2015), 3781-3790 Request permission
Abstract:
The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where the $F^*_i(G)$ is the generalized Fitting series: $F^*_1(G)=F^*(G)$ and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if $G$ admits a soluble group of automorphisms $A$ of coprime order, then $h^*(G)$ is bounded in terms of $h^* (C_G(A))$, where $C_G(A)$ is the fixed-point subgroup, and the number of prime factors of $|A|$ counting multiplicities. The result follows from the special case when $A=\langle \varphi \rangle$ is of prime order, where it is proved that $F^*(C_G(\varphi ))\leqslant F^*_{9}(G)$.
The nonsoluble length $\lambda (G)$ of a finite group $G$ 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 $A$ is a group of automorphisms of $G$ of coprime order, then $\lambda (G)$ is bounded in terms of $\lambda (C_G(A))$ and the number of prime factors of $|A|$ counting multiplicities.
References
- P. Hall and Graham Higman, On the $p$-length of $p$-soluble groups and reduction theorems for Burnside’s problem, Proc. London Math. Soc. (3) 6 (1956), 1–42. MR 72872, DOI 10.1112/plms/s3-6.1.1
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
- Bertram Huppert and Norman Blackburn, Finite groups. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 243, Springer-Verlag, Berlin-New York, 1982. MR 662826
- E. I. Khukhro and P. Shumyatsky, Nonsoluble and non-$p$-soluble length of finite groups, to appear in Israel J. Math.; \verb#arXiv:1310.2434#.
- 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, DOI 10.1017/S1446788714000317
- Hans Kurzweil, $p$-Automorphismen von auflösbaren $p^{\prime }$-Gruppen, Math. Z. 120 (1971), 326–354 (German). MR 284503, DOI 10.1007/BF01109999
- 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 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
- 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
- John S. Wilson, On the structure of compact torsion groups, Monatsh. Math. 96 (1983), no. 1, 57–66. MR 721596, DOI 10.1007/BF01298934
Additional Information
- E. I. Khukhro
- Affiliation: Sobolev Institute of Mathematics, Novosibirsk, 630 090, Russia – and – University of Lincoln, Lincoln, United Kingdom.
- MR Author ID: 227765
- Email: khukhro@yahoo.co.uk
- P. Shumyatsky
- Affiliation: Department of Mathematics, University of Brasilia, DF 70910-900, Brazil
- MR Author ID: 250501
- Email: pavel@unb.br
- 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 CNPq-Brazil. The first author thanks CNPq-Brazil and the University of Brasilia for support and hospitality that he enjoyed during his visits to Brasilia.
- Communicated by: Pham Huu Tiep
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3781-3790
- MSC (2000): Primary 20D45
- DOI: https://doi.org/10.1090/S0002-9939-2015-12573-7
- MathSciNet review: 3359570