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Extending the Kegel Wielandt theorem through π-decomposable groups

Published online by Cambridge University Press:  05 July 2011

By
L. S. Kazarin ,
A. Martínez-Pastor  and
M. D. Pérez-Ramos
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
L. S. Kazarin
Affiliation:
Yaroslavl P. Demidov State University
A. Martínez-Pastor
Affiliation:
Universidad Politécnica de Valencia
M. D. Pérez-Ramos
Affiliation:
Universitat de València
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

A celebrated theorem of Kegel and Wielandt asserts the solubility of a finite group which is the product of two nilpotent subgroups. In this survey we report on some extensions of this result by considering π-decomposable subgroups, for a set of primes π, instead of nilpotent groups.

Introduction

The study of groups which can be factorised as the product of two subgroups has developed extensively in recent decades. The general aim is to obtain information about the structure of the whole group from the structure of the subgroups in the factorization, and vice versa. An example is the well known theorem of Kegel and Wielandt which establishes the solubility of a finite group factorised as the product of two nilpotent subgroups. This result has been the motivation for a wide variety of results in the literature. In particular some of them consider the situation when either one or both of the factors are π-decomposable, for a set of primes π. This paper is a survey article containing a detailed account of recent achievements which extend the Kegel–Wielandt theorem in this direction.

Only finite groups are considered in this paper.

Let us start with an explicit statement of the starting point of our development:

Theorem 1.1 (Kegel [14] and Wielandt [19])If the group G = AB is the product of two nilpotent subgroups A and B, then G is soluble.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 415 - 423
Publisher: Cambridge University Press
Print publication year: 2011

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References

[1] B., Amberg, S., Franciosi and F., de Giovanni, Products of Groups (Clarendon Press, Oxford 1992).Google Scholar
[2] Z., Arad and D., Chillag, Finite groups containing a nilpotent Hall subgroup of even order, Houston J. Math. 7 (1981), 23–32.Google Scholar
[3] Z., Arad and E., Fisman, On finite factorizable groups, J. Algebra 86 (1984), 522–548.Google Scholar
[4] Ya. G., Berkovich, Generalization of the theorems of Carter and Wielandt, Sov. Math. Dokl. 7 (1966), 1525–1529.Google Scholar
[5] R., Carter, Simple groups of Lie type (Wiley, London 1972).Google Scholar
[6] J. H., Conway, R. T., Curtis, S. P., Norton, R. A., Parker and R. A., Wilson, Atlas of Finite Groups (OUP, Oxford 1985). http://brauer.maths.qmul.ac.uk/Atlas/v3/Google Scholar
[7] E., Fisman, On the product of two finite solvable groups, J. Algebra 80 (1983), 517–536.Google Scholar
[8] R., Guralnick, Subgroups of prime power index in a simple group, J. Algebra 81 (1983), 304–311.Google Scholar
[9] L. S., Kazarin, Criteria for the nonsimplicity of factorable groups, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 288–308.Google Scholar
[10] L. S., Kazarin, The product of a 2-decomposable group and a group of odd order, Problems in group theory and homological algebra (1983), 89–98. (Russian)Google Scholar
[11] L. S., Kazarin, A., Martínez-Pastor and M. D., Pérez-Ramos, On the product of a π-group and a π-decomposable group, J. Algebra 315 (2007), 640–653.Google Scholar
[12] L. S., Kazarin, A., Martínez-Pastor and M. D., Pérez-Ramos, On the product of two π-decomposable soluble groups, Publ. Mat. 53 (2) (2009), 439–456.Google Scholar
[13] L. S., Kazarin, A., Martínez-Pastor and M. D., Pérez-Ramos, On the product of two π-decomposable groups, preprint.
[14] O. H., Kegel, Produkte nilpotenter Gruppen, Arch. Math. 12 (1961), 90–93.Google Scholar
[15] A. S., Kondratiev and V. D., Mazurov, 2-signalizers of finite simple groups, Algebra Logic 42 (2003), 333–348.Google Scholar
[16] M., Liebeck, C. E., Praeger, and J., Saxl, The maximal factorizations of the finite simple groups and their automorphism groups, Mem. Amer. Math. Soc. 86, No. 432, (Amer. Math. Soc., Providence, RI, 1990).Google Scholar
[17] V. S., Monakhov, Solvability of a factorable group with decomposable factors, Mat. Zametki 34 (1983), 337–340. (Russian)Google Scholar
[18] P. J., Rowley, The π-separability of certain factorizable groups, Math. Z. 153 (1977), 219–228.Google Scholar
[19] H., Wielandt, Über Produkte von nilpotenten Gruppen, Ill. J. Math. 2 (1958), 611–618.Google Scholar
[20] H., Wielandt, Vertauschbarkeit von Untergruppen und Subnormalität, Math. Z. 133 (1973), 275–276.Google Scholar

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