Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T08:59:29.074Z Has data issue: false hasContentIssue false
  • English
  • Français

CLASSIFICATION OF FINITE GROUPS VIA THEIR BREADTH

Part of: Representation theory of groups

Published online by Cambridge University Press:  27 June 2019

HERMANN HEINEKEN  and
FRANCESCO G. RUSSO Open the ORCID record for FRANCESCO G. RUSSO [Opens in a new window]
HERMANN HEINEKEN
Affiliation:
Department of Mathematics, University of Würzburg, Campus Hubland Nord, Email-Fischer-Strasse 30 97074, Würzburg, Germany e-mail: [email protected]
FRANCESCO G. RUSSO
Affiliation:
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X1, Rondebosch 7701, Cape Town, South Africa e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let k be a divisor of a finite group G and Lk(G) = {x ∈ G | xk =1}. Frobenius proved that the number |Lk(G)| is always divisible by k. The following inverse problem is considered: for a given integer n, find all groups G such that max{k-1|Lk(G)| | k ∈ Div(G)} = n, where Div(G) denotes the set of all divisors of |G|. A procedure beginning with (in a sense) minimal members and deducing the remaining ones is outlined and executed for n=8.

MSC classification

Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Secondary: 20D15: Nilpotent groups, $p$-groups 20D60: Arithmetic and combinatorial problems
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 3 , September 2020 , pp. 544 - 563
Copyright
© Glasgow Mathematical Journal Trust 2019

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

Berkovich, Y. and Janko, Z., Groups of prime power order, Vols. 1, 2, 3 (de Gruyter, Berlin, 2010, 2010, 2011).Google Scholar
Doerk, K. and Hawkes, T., Finite Soluble Groups (de Gruyter, Berlin, 1992).CrossRefGoogle Scholar
Frobenius, G., Verallgemeinerung des Sylowschen Satzen, Berliner Sitz. (1895), 981–993.Google Scholar
Frobenius, G., Über einen Fundamentalsatz der Gruppentheorie, Berliner Sitz. (1903), 987–991.Google Scholar
Heineken, H. and Russo, F. G., Groups described by element numbers, Forum Math. 27 (2015), 1961–1977.CrossRefGoogle Scholar
Heineken, H. and Russo, F. G., On a notion of breadth in the sense of Frobenius, J. Algebra 424 (2015), 208–221.CrossRefGoogle Scholar
Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
Jaikin-Zapirain, A. and Pyber, L., Random generation of finite and profinite groups and group enumeration, Annals Math. 173 (2011), 769–814.CrossRefGoogle Scholar
Meng, W. and Shi, J., On an inverse problem to Frobenius’ theorem, Arch. Math. (Basel) 96 (2011), 109–114.CrossRefGoogle Scholar
Meng, W., Shi, J. and Chen, K., On an inverse problem to Frobenius’ theorem II, J. Algebra Appl. 11 (2012), Paper ID: 1250092.CrossRefGoogle Scholar
Meng, W., Finite groups of global breadth four in the sense of Frobenius, Comm. Algebra 45 (2016), 660–665.CrossRefGoogle Scholar
Shi, J., Meng, W. and Zhang, C., On the Frobenius spectrum of a finite group, J. Algebra Appl. 16 (2017), Paper ID: 1750051.CrossRefGoogle Scholar