通过基于子群共轭类的不变量检测可解性、超可解性和CLT性质
Detecting solvability, supersolvability and CLT properties via an invariant based on conjugacy classes of subgroups
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中文总结 AI 辅助
该研究通过定义基于子群共轭类的不变量d*(G)，推导得出d*(G)与有限群G的可解性、超可解性及CLT性质相关的判定准则。
中文摘要 AI 辅助
对于有限群G，记k'(G)为子群共轭类的数量，L(G)为G的子群格，令d'(G)=k'(G)/|L(G)|，d*(G)为G的所有截面S的d'(S)的最小值。本文推导了关于G性质的若干判据：若d*(G)>9/59，则G可解；若d*(G)>1/2，则G是超可解群；将“超可解”替换为“CLT”时，该判据仍成立。
英文摘要
For a finite group $G$, denote by $k'(G)$ and $L(G)$ the number of conjugacy classes of subgroups and the subgroup lattice of $G$, respectively. Let $d'(G)=\frac{k'(G)}{|L(G)|}$ and $d^*(G)$ be the minimum value of $d'(S)$, when $S$ runs through all sections of $G$. In this paper we deduce some criteria on the nature of $G$. We show that if $d^*(G)>\frac{9}{59}$, then $G$ is solvable, while if $d^*(G)>\frac{1}{2}$, then $G$ is a supersolvable group. The last criterion is also valid when replacing ``supersolvable" with ``CLT".
