A Note on the Normal Index and the c-Section of Maximal Subgroups of a Finite Group

Abstract

Let $M$ be a maximal subgroup of finite group $G$. For each chief factor $H/K$ of $G$ such that $K\le M$ and $G=MH$, we called the order of $H/K$ the normal index of $M$ and $\left(M\cap H\right)/K$ a section of $M$ in $G$. Using the concepts of normal index and c-section, we obtain some new characterizations of p-solvable, 2-supersolvable, and p-nilpotent.