Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-15T23:55:56.006Z Has data issue: false hasContentIssue false
  • English
  • Français

SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p-ELEMENTS

Part of: Representation theory of groups

Published online by Cambridge University Press:  26 April 2010

GIL KAPLAN  and
DAN LEVY
GIL KAPLAN*
Affiliation:
The School of Computer Sciences, The Academic College of Tel-Aviv-Yaffo, 2 Rabenu Yeruham St., Tel-Aviv 61083, Israel (email: [email protected])
DAN LEVY
Affiliation:
The School of Computer Sciences, The Academic College of Tel-Aviv-Yaffo, 2 Rabenu Yeruham St., Tel-Aviv 61083, Israel
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc.12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc.12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc.74(3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p′-elements, a formation Pp,p′, for each prime p. We show that P2,2′ (which contains the odd-order groups) is properly contained in the solvable formation.

Keywords

solvable groupsconjugate elementsformations

MSC classification

Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 2 , October 2010 , pp. 265 - 273
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Barry, M. J. J. and Ward, M. B., ‘Simple groups contain minimal simple groups’, Publicacions Matemàtiques 41 (1997), 411–415.CrossRefGoogle Scholar
[2]Collins, M. J., Representations and Characters of Finite Groups (Cambridge University Press, Cambridge, 1990).Google Scholar
[3]Dixon, J. D. and Mortimer, B., Permutation Groups, Graduate Texts in Mathematics, 163 (Springer, Berlin, 1996).Google Scholar
[4]Doerk, K. and Hawkes, T., Finite Soluble Groups, de Gruyter Expositions in Mathematics, 4 (de Gruyter, Berlin, 1992).CrossRefGoogle Scholar
[5]Flavell, P., ‘Finite groups in which every two elements generate a soluble group’, Invent. Math. 121 (1995), 279–285.CrossRefGoogle Scholar
[6]Gehles, K. E., ‘Ordinary characters of finite special linear groups’, MSc Dissertation, School of Mathematics and Statistics, University of St. Andrews, 2002. Downloadable from http://www-circa.mcs.st-and.ac.uk/theses.html#msc .Google Scholar
[7]Gordeev, N., Grunewald, F., Kunyavskii, B. and Plotkin, E., ‘Baer–Suzuki theorem for the solvable radical of a finite group’, Comptes Rendus Acad. Sci. Paris, Ser. I 347 (2009), 217–222.CrossRefGoogle Scholar
[8]Guest, S., ‘A solvable version of the Baer–Suzuki theorem’, Trans. Amer. Math. Soc. in press.Google Scholar
[9]Hall, P., ‘A characteristic property of soluble groups’, J. London Math. Soc. 12 (1937), 198–200.Google Scholar
[10]Kurzweil, H. and Stellmacher, B., The Theory of Finite Groups: An Introduction (Springer, Berlin, 2004).Google Scholar
[11]Miller, G., ‘The product of two or more groups’, Trans. Amer. Math. Soc. 12 (1911), 326.CrossRefGoogle Scholar
[12]Suzuki, M., ‘On a class of doubly transitive groups II’, Ann. of Math. (2) 79 (1964), 514–589.CrossRefGoogle Scholar
[13] The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.4.12 of 17-Dec-2008 (http://www.gap-system.org).Google Scholar
[14]Thompson, J. G., ‘Nonsolvable finite groups all of whose local subgroups are solvable’, Bull. Amer. Math. Soc. 74(3) (1968), 383–437.CrossRefGoogle Scholar