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<p>The order sequence of a finite group <i>G</i> is a non-decreasing finite sequence formed of the element orders of <i>G</i>. Several properties of order sequences were studied by P. J. Cameron and H. K. Dey in a recent paper that concludes with a list of open problems. In this paper, we solve two of these problems by showing the following facts: 1) If there is a non-supersolvable/non-solvable group of order <i>n</i>, it is not always true that its order sequence is properly dominated by the order sequence of any supersolvable/solvable group of order <i>n</i>; 2) the supersolvability of a finite group cannot be described by its order sequence.</p>

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On two problems about order sequences of finite groups

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  • 发表日期：

    2025年9月3日

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  • Mihai-Silviu Lazorec

摘要

The order sequence of a finite group G is a non-decreasing finite sequence formed of the element orders of G. Several properties of order sequences were studied by P. J. Cameron and H. K. Dey in a recent paper that concludes with a list of open problems. In this paper, we solve two of these problems by showing the following facts: 1) If there is a non-supersolvable/non-solvable group of order n, it is not always true that its order sequence is properly dominated by the order sequence of any supersolvable/solvable group of order n; 2) the supersolvability of a finite group cannot be described by its order sequence.