Supersolvable subgroups of order divisible by 3

Abstract

We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying the above condition on maximal subgroups, then either $G$ is a $3^{\prime }$-group or $G/{\mathbf{O}}_{3^{\prime }}(G)$ is isomorphic to ${\mathrm{PSL}}_2(2^p)$ for an odd prime $p$, where ${\mathbf{O}}_{3^{\prime }}(G)$ denotes the largest normal $3^{\prime }$-subgroup of $G$. Furthermore, in the latter case, ${\mathbf{O}}_{3^{\prime }}(G)$ is nilpotent and ${\mathbf{O}}_2(G)\le \mathbf{Z}(G)$.