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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On large cyclic subgroups of finite groups
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by Edward A. Bertram PDF
Proc. Amer. Math. Soc. 56 (1976), 63-66 Request permission

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 $G$.
References
Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 56 (1976), 63-66
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0399019-5
  • MathSciNet review: 0399019