Abstract
In this paper, we study the ratio between the number of p-elements and the order of a Sylow p-subgroup of a finite group G. As well known, this ratio is a positive integer and we conjecture that, for every group G, it is at least the \((1-\frac{1}{p})\)-th power of the number of Sylow p-subgroups of G. We prove this conjecture if G is p-solvable. Moreover, we prove that the conjecture is true in its generality if a somewhat similar condition holds for every almost simple group.
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Acknowledgements
This article is part of the author’s PhD thesis, which was written under the great supervision of Carlo Casolo, whose contribution to this work was essential. Thanks are also due to Francesco Fumagalli and Silvio Dolfi for his valuable comments and suggestions. This work was partially funded by the Istituto Nazionale di Alta Matematica “Francesco Severi” (Indam).
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This work is dedicated to the memory of Carlo Casolo. His knowledge, his curiosity, his humility and his humanity were an example to all of his students and friends.
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Gheri, P. On the number of p-elements in a finite group. Annali di Matematica 200, 1231–1243 (2021). https://doi.org/10.1007/s10231-020-01035-9
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DOI: https://doi.org/10.1007/s10231-020-01035-9