Skip to main content Accessibility help
×
Hostname: page-component-788cddb947-jbjwg Total loading time: 0 Render date: 2024-10-08T10:19:37.006Z Has data issue: false hasContentIssue false

Miscellaneous results on supersolvable groups

Published online by Cambridge University Press:  05 July 2011

K. Corrádi
Affiliation:
Eötvös Loránd University, Hungary
P. Z. Hermann
Affiliation:
Eötvös Loránd University, Hungary
L. Héthelyi
Affiliation:
Budapest University of Technology and Economics, Hungary
E. Horváth
Affiliation:
Budapest University of Technology and Economics, Hungary
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
Get access

Summary

Abstract

The paper contains two theorems generalizing the theorems of Huppert concerning the characterization of supersolvable and p-supersolvable groups, respectively. The first of these gives a new approach to prove Huppert's first named result. The second one has numerous applications in the paper. The notion of balanced pairs is introduced for non-conjugate maximal subgroups of a finite group. By means of them some new deep results are proved that ensure supersolvability of a finite group.

Introduction

We recall Huppert's characterizations for (p-)supersolvable groups.

  1. (i) Let p be some prime. A finite group is p-supersolvable iff it is p-solvable and the index of any maximal subgroup is either p or coprime to p.

  2. (ii) A finite group is supersolvable iff all maximal subgroups of it have prime index.

(See in [10, 9.2–9.5 Satz], pp. 717–718.) Among others it immediately follows that the class (formation) of finite supersolvable groups is saturated, i.e. the supersolvability of G/Φ(G) is equivalent to the supersolvability of G itself. Result (ii) turned out to be of fundamental importance and it inspired a long series of further achievements. Concentrating to various characterizations of finite supersolvable groups by means of the index of maximal subgroups or the existence of cyclic supplements to maximal subgroups we mention [7], [12] and [15] from the past; cf. also [16] (or [6, Thm. 2.2], p 483).

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] W., Burnside, On groups of order pa qb, Proc. London Math. Soc. 2 (1904) 388–392.Google Scholar
[2] Chen, Shun, Min & Zhang, Liang, Cai, Some results on supersolvable groups (Chinese), J. Nanjing Norm. Univ. Nat. Sci. Ed. 28 (2005), no. 2, 33–37.Google Scholar
[3] K., Corrádi, Problem at the Schweitzer Competition, Mat. Lapok 20 (1969) 159–162.Google Scholar
[4] P., Csörgő, On the natural factorization of finite supersolvable groups, in Groups–Korea '98 (Pusan), 91–94, de Gruyter, Berlin, 2000.Google Scholar
[5] P., Csörgő, On supersolvability of finite groups, Glasgow Math. J. 43 (2001) 327–333.Google Scholar
[6] K., Doerk & T., Hawkes, Finite soluble groups, de Gruyter Expositions in Mathematics, 4, Berlin, 1992Google Scholar
[7] T. K., Dutta, A., Bhattacharyya, Some results on π-solvable and supersolvable groups, Internat. J. Math. Sci. 17 (1994), no. 1, 59–64.Google Scholar
[8] Guo, Xiuyun & K. P., Shum, On finite supersolvable groups and saturated formations, Int. Math. J. 1 (2002), no. 6, 621–630.Google Scholar
[9] P., Hall, A characteristic property of soluble groups, J. London Math. Soc. 12 (1937) 188–200.Google Scholar
[10] B., Huppert, Endliche Gruppen I., Springer, 1967.Google Scholar
[11] L., Lovász, Combinatorial problems and exercises, Akadémiai Kiadó Budapest and North-Holland Publishing Company, 1979.Google Scholar
[12] N. P., Mukherjee & Prabir, Bhattacharya, On supersolvable groups and a theorem of Huppert, Canad. Math. Bull. 33 (1990), no. 3, 314–315.Google Scholar
[13] L., Rédei, Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen 4 (1956) 303–324.Google Scholar
[14] O. L., Shemetkova, On the Vedernikov–Kuleshov theorem for finite supersolvable groups (Russian), Dokl. Akad. Nauk 396 (2004), no. 5, 608–610.Google Scholar
[15] V. A., Vedernikov & N. I., Kuleshov, Characterization of finite supersolvable groups (Russian, English summary), in Problems in algebra, No. 9 (Russian), 107–113, Gomel. Gos. Univ., Gomel, 1996.Google Scholar
[16] P., Venzke, Maximal subgroups of prime index in a finite solvable group, Proc. Amer. Math. Soc. 68 (1978), no. 2, 140–142.Google Scholar
[17] Wang, Kun, Ren, Some necessary and sufficient conditions for finite supersolvable groups II (Chinese), Sichuan Shifan Daxue Xuebao Ziran Kexue Ban 26 (2003), no. 5, 445–447.Google Scholar
[18] H., Wielandt, Zum Satz von Sylow, Math. Z. 60 (1954), 407–409.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×