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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On large cyclic subgroups of finite groups


Author: Edward A. Bertram
Journal: Proc. Amer. Math. Soc. 56 (1976), 63-66
MathSciNet review: 0399019
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Abstract: It is known that for each (composite) $ n$ every group of order $ n$ contains a proper subgroup of order greater than $ {n^{1/3}}$. We prove that given $ 0 < \delta < 1$, for almost all $ n \leqslant x$, as $ x \to \infty $, every group $ G$ of order $ n$ contains a characteristic cyclic subgroup of square-free order $ > {n^{1 - 1/{{(\log n)}^{1 - \delta }}}}$, and provide an upper bound to the number of exceptional $ n$. This leads immediately to a like density result for a lower bound to the number of conjugacy classes in $ G$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0399019-5
Article copyright: © Copyright 1976 American Mathematical Society