Abstract
Let G be a finite group and let \(p_1,\dots ,p_n\) be distinct primes. If G contains an element of order \(p_1 \cdots p_n,\) then there is an element in G which is not contained in the Frattini subgroup of G and whose order is divisible by \(p_1 \cdots p_n.\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. D. Dixon, Random sets which invariably generate the symmetric group, Discrete Math. 105 (1992), 25–39.
R. Guralnick, W. Kantor, M. Kassabov, and A. Lubotzky, Presentations of finite simple groups: profinite and cohomological approaches, Groups Geom. Dyn. 1 (2007), 469–523.
B. Huppert, Endliche Gruppen, Grundlehren der Mathematischen Wissenschaften, 134, Springer-Verlag, Berlin-New York, 1967.
W. M. Kantor, A. Lubotzky, and A. Shalev, Invariable generation and the Chebotarev invariant of a finite group, J. Algebra 348 (2011), 302–314.
A.S. Kondrat’ev, On prime graph components of finite simple groups, Math. USSR-Sb. 67 (1990), 235–247.
L. G. Kovács, J. Neubüser, and B. H. Neumann, On finite groups with “hidden” primes, J. Austral. Math. Soc. 12 (1971), 287–300.
D. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York, 1993.
J.S. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), 487–513.
A.V. Zavarnitsine, On the recognition of finite groups by the prime graph, Algebra Logic 45 (2006), 220–231.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lucchini, A. On the orders of the non-Frattini elements of a finite group. Arch. Math. 110, 421–426 (2018). https://doi.org/10.1007/s00013-017-1147-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-017-1147-8