Abstract
Let G be a finite group and π be a set of primes. Put \({d_{\pi}(G) = k_{\pi}(G)/|G|_{\pi}}\), where \({k_{\pi}(G)}\) is the number of conjugacy classes of π-elements in G and |G| π is the π-part of the order of G. In this paper we initiate the study of this invariant by showing that if \({d_{\pi}(G) > 5/8}\) then G possesses an abelian Hall π-subgroup, all Hall π-subgroups of G are conjugate, and every π-subgroup of G lies in some Hall π-subgroup of G. Furthermore, we have \({d_{\pi}(G) = 1}\) or \({d_{\pi}(G) = 2/3}\). This extends and generalizes a result of W. H. Gustafson.
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The research of the first author was supported by a Marie Curie International Reintegration Grant within the 7th European Community Framework Programme, by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by an Alexander von Humboldt Fellowship for Experienced Researchers, by OTKA K84233, and by the MTA RAMKI Lendület Cryptography Research Group.
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Maróti, A., Nguyen, H.N. On the number of conjugacy classes of π-elements in finite groups. Arch. Math. 102, 101–108 (2014). https://doi.org/10.1007/s00013-014-0615-7
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DOI: https://doi.org/10.1007/s00013-014-0615-7