Abstract
Every finite group G has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length λ(G) as the number of nonsoluble factors in a shortest series of this kind. Upper bounds for λ(G) appear in the study of various problems on finite, residually finite, and profinite groups. We prove that λ(G) is bounded in terms of the maximum 2-length of soluble subgroups of G, and that λ(G) is bounded by the maximum Fitting height of soluble subgroups. For an odd prime p, the non-p-soluble length λ p (G) is introduced, and it is proved that λ p (G) does not exceed the maximum p-length of p-soluble subgroups. We conjecture that for a given prime p and a given proper group variety V the non-p-soluble length λ p (G) of finite groups G whose Sylow p-subgroups belong to V is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent. As an application of the results obtained, an error is corrected in the proof of the main result of the second author’s paper Multilinear commutators in residually finite groups, Israel Journal of Mathematics 189 (2012), 207–224.
Similar content being viewed by others
References
T. R. Berger and F. Gross, 2-length and the derived length of a Sylow 2-subgroup, Proceedings of the London Mathematical Society 34 (1977), 520–534.
E. G. Bryukhanova, The 2-length and 2-period of a finite solvable group, Algebra i Logika 18 (1979), 9–31; English translation: Algebra and Logic 18 (1979), 5–20.
E. G. Bryukhanova, The relation between 2-length and derived length of a Sylow 2-subgroup of a finite soluble group, Matematicheskie Zametki 29 (1981), no. 2, 161–170; English translation: Mathematical Notes 29 (1981), 85–90.
E. Detomi, M. Morigi and P. Shumyatsky, Bounding the exponent of a verbal subgroup, Annali Mat., doi:10.1007/s10231-013-0336-8, to appear.
W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific Journal of Mathematics 13 (1963), 773–1029.
D. Gorenstein, Finite Groups, Chelsea Publishing Company, New York, 1980.
F. Gross, The 2-length of a finite solvable group, Pacific Journal of Mathematics 15 (1965), 1221–1237.
M. Hall, The Theory of Groups, Macmillan, New York, 1959.
P. Hall and G. Higman, The p-length of a p-soluble group and reduction theorems for Burnside’s problem, Proceedings of the London Mathematical Society 6 (1956), 1–42.
A. H. M. Hoare, A note on 2-soluble groups, Journal of the London Mathematical Society 35 (1960), 193–199.
G. A. Jones, Varieties and simple groups, Journal of the Australian Mathematical Society 17 (1974), 163–173.
P. Shumyatsky, Commutators in residually finite groups, Israel Journal of Mathematics 182 (2011), 149–156.
P. Shumyatsky, Multilinear commutators in residually finite groups, Israel Journal of Mathematics 189 (2012), 207–224.
P. Shumyatsky, On the exponent of a verbal subgroup in a finite group, Journal of the Australian Mathematical Society 93 (2012), 325–332.
Unsolved Problems in Group Theory, The Kourovka Notebook, no. 17, Institute of Mathematics, Novosibirsk, 2010.
J. Wilson, On the structure of compact torsion groups, Monatshefte für Mathematik 96, 57–66.
E. I. Zelmanov, On some problems of the theory of groups and Lie algebras, Matematicheski ĭ Sbornik 180 (1989), 159–167; English translation: Mathematics of the USSR-Sbornik 66 (1990), 159–168.
E. I. Zelmanov, A solution of the Restricted Burnside Problem for groups of odd exponent, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 54 (1990), 42–59; English translation: Mathematics of the USSR-Izvestiya 36 (1991), 41–60.
E. I. Zelmanov, A solution of the Restricted Burnside Problem for 2-groups, Matematicheski ĭ Sbornik 182 (1991), 568–592; English translation: Mathematics of the USSRSbornik 72 (1992), 543–565.
E. I. Zelmanov, On periodic compact groups, Israel Journal of Mathematics 77 (1992), 83–95.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
Khukhro, E.I., Shumyatsky, P. Nonsoluble and non-p-soluble length of finite groups. Isr. J. Math. 207, 507–525 (2015). https://doi.org/10.1007/s11856-015-1180-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-015-1180-x