有限群的循环概率
Cyclic Probability of a Finite Group
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中文总结 AI 辅助
研究有限群\(G\)的循环概率\(\theta(G)\)，通过研究其上下界，表明它可用于检验群的幂零性与可解性，还与交换概率等其他群不变量作比较。
中文摘要 AI 辅助
有限群\(G\)的循环概率\(\theta(G)\)定义为\(G\)中两个随机选取的元素生成\(G\)的循环子群的概率。本文研究了\(\theta(G)\)的各种上下界，表明\(\theta(G)\)可作为检验有限群幂零性和可解性的标准。还将\(\theta(G)\)与其他群不变量如交换概率\(cp(G)\)和元素阶的归一化和进行了比较。
英文摘要
The cyclic probability, $θ(G)$, of a finite group $G$, is defined as the probability that two randomly selected elements of $G$ generate a cyclic subgroup of $G$. In this paper, we study various upper and lower bounds of $θ(G)$ and show that $θ(G)$ can be used as a criterion for checking nilpotency and solvability of a finite group. We also compare $θ(G)$ with other group invariants like commuting probability $cp(G)$ and normalized sum of element orders.
