×

An infinitude of counterexamples to Herzog’s conjecture on involutions in simple groups. (English) Zbl 1506.20056

Summary: In 1979, M. Herzog [Proc. Am. Math. Soc. 77, 313–314 (1979; Zbl 0421.20009)] conjectured that two finite simple groups containing the same number of involutions have the same order. M. Zarrin, in a 2018 published paper [Arch. Math. 111, No. 4, 349–351 (2018; Zbl 1425.20017)], disproved Herzog’s conjecture with a counterexample. The goal of this article is to prove that there are infinitely many counterexamples to Herzog’s conjecture. In doing so, we obtain an explicit formula for the number of involutions in the groups involved.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D06 Simple groups: alternating groups and groups of Lie type
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anabanti, C. S., A counterexample to Zarrin’s Conjecture on sizes of finite nonabelian simple groups in relation to involution sizes, Arch. Math, 112, 3, 225-226 (2019) · Zbl 1446.20036 · doi:10.1007/s00013-018-1265-y
[2] Carter, R. W., Simple Groups of Lie Type, 364 (1972), New York: Wiley Classics Library, John Wiley & Sons Ltd, New York · Zbl 0248.20015
[3] Fulman, J.; Guralnick, R.; Stanton, D., Asymptotics of the number of involutions in finite classical groups, J. Group Theory, 20, 5, 871-902 (2017) · Zbl 1376.20055 · doi:10.1515/jgth-2017-0011
[4] Herzog, M., On the classification of finite simple groups by the number of involutions, Proc. Am. Math. Soc, 77, 3, 313-314 (1979) · Zbl 0421.20009 · doi:10.1090/S0002-9939-1979-0545587-2
[5] Taghvasani, L. J.; Zarrin, M., A characterization of A_5 by its same-order type, Monatsh. Math, 182, 3, 731-736 (2017) · Zbl 1364.20007 · doi:10.1007/s00605-016-0950-9
[6] Zarrin, M., A counterexample to Herzog’s Conjecture on the number of involutions, Arch. Math, 111, 4, 349-351 (2018) · Zbl 1425.20017 · doi:10.1007/s00013-018-1195-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.