On a bijection between a finite group to a non-cyclic group with divisibility of element orders

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发表日期：

2025年2月13日

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Mohsen Amiri

摘要

Consider a finite group G of order n with a prime divisor p. In this article, we establish, among other results, that if the Sylow p-subgroup of G is neither cyclic nor generalized quaternion, then there exists a bijection f from G onto the abelian group $C_{\frac{n}{p}}\times C_p$ $C_{\frac{n}{p}} \times C_{p}$ such that for every element x in G, the order of x divides the order of f(x). This resolves Question 1.5 posed in Jafarian Amiri and Amiri (Comm. Algebra 45:3396-3401, 2017). As application of our results, we show that the group with the third largest value of the sum of element orders in the set of all finite groups of order n is a solvable p-nilpotent group where p is the smallest prime divisor of n such that the Sylow p-subgroups are not cyclic.