元素阶和函数的一个推广
A generalization of the sum of element orders function
- Presidency University（普雷西登西大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对有限群引入元素阶和函数的推广形式，得到其关于循环、交换、幂零及可解群的阈值，且该阈值优于其他参数对应的阈值，还研究了该函数的相关性质。
AI中文摘要：
本文针对有限群$G$和正整数$k$，引入函数$$\psi''_k(G)=\dfrac{\psi_k(G)}{|G|^{k+1}}, \quad \text{where} \quad \psi_k(G)=\sum_{g \in G}o(g)^k,$$，它是M. T{ă}rn{ă}uceanu提出的函数$\psi''$的第$k$个推广形式。得到了循环群、交换群、幂零群及（超）可解群关于$k$的不同阈值，结果表明这些阈值通常优于从$\psi'$和$\psi''$等其他参数导出的类似阈值。此外，将$\psi''_k$视为$(0,1]$值函数，研究其单射性、满射性和稠密性。
英文摘要：
In this paper, for a finite group $G$ and a positive integer $k$, we introduce the function $$ψ''_k(G)=\dfrac{ψ_k(G)}{|G|^{k+1}}, \quad \text{where} \quad ψ_k(G)=\sum_{g \in G}o(g)^k,$$ as the $k$-th generalization of the function $ψ''$, introduced by M. T{ă}rn{ă}uceanu. Different thresholds for cyclic, abelian, nilpotent and (super)solvable groups in terms of $k$ are obtained. It is shown that these thresholds often outperform similar thresholds derived from other parameters like $ψ'$ and $ψ''$. Moreover, considering $ψ''_k$ as a $(0,1]$-valued function, we study its injectivity, surjectivity and density.
