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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: heineken@mathematik.uni-wuerzburg.de
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: francescog.russo@yahoo.com
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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

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