融合系统超可解性与有限群$p$-幂零性的新刻画
New characterizations for supersolvability of fusion system and $p$-nilpotency of finite groups
AI总结：
本文在子群满足广义正规性的条件下，给出了融合系统超可解性及有限群$p$-幂零性的新刻画，并推广至广义饱和融合系统。
摘要：
设$p$为素数，$S$为$p$-群，$\mathcal{F}$为$S$上的饱和融合系统。若存在$S$的一个序列$1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$，使得$S_{i+1}/S_i$为循环群，$i=0,1,\cdots, n-1$，且$S_i$是强$\mathcal{F}$-闭的，$i=0,1,\cdots,n$，则称$\mathcal{F}$是超可解的。本文在$G$的某些子群满足不同种类的广义正规性的假设下，研究了$\mathcal{F}_S (G)$的超可解性的刻画。此外，我们在第4节中获得了关于广义饱和融合系统$\mathcal{F}$的刻画这一更先进且显著的结果。最后，我们应用第3节和第4节的结果，在$G$的某些子群满足不同种类的广义正规性的假设下，给出了有限群$p$-幂零性的刻画。
英文摘要：
Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, $i=0,1,\cdots, n-1$, $S_i$ is strongly $\mathcal{F}$-closed, $i=0,1,\cdots,n$. In this paper, we investigate the characterizations for supersolvability of $\mathcal{F}_S (G)$ under the assumption that certain subgroups of $G$ satisfy different kinds of generalized normalities in section \ref{1003}. Moreover, we obtain the more advanced and remarkable result of characterizations for generalized saturated fusion system $\mathcal{F}$ in section \ref{Section 4}. Finally, we apply the results in section \ref{1003} and \ref{Section 4} and give characterizations for $p$-nilpotency of finite groups under the assumption that some subgroups of $G$ satisfy different kinds of generalized normalities.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
