A Note on the Solvablity of Groups

Shiheng Li, Wujie Shi

Published 2005-09-16, updated 2005-09-21Version 2

Let $M$ be a maximal subgroup of a finite group $G$ and $K/L$ be a chief factor such that $L\leq M$ while $K\nsubseteq M$. We call the group $M\cap K/L$ a $c$\ns section of $M$. And we define $Sec(M)$ to be the abstract group that is isomorphic to a $c$\ns section of $M$. For every maximal subgroup $M$ of $G$, assume that Sec($M$) is supersolvable. Then any composition factor of $G$ is isomorphic to $L_2(p)$ or $Z_q$, where $p$ and $q$ are primes, and $p\equiv\pm 1(mod 8)$. This result answer a question posed by ref. \cite{WL}.