Supersolvable subgroups of order divisible by 3
Antonio Beltrán, Changguo Shao
TL;DR
This work classifies finite non-solvable groups with order divisible by $3$ in which every maximal subgroup of order divisible by $3$ is supersolvable. Using a minimal-simple-group framework and the Classification of Finite Simple Groups, the authors show that such a group $G$ satisfies either being a $3'$-group or having a quotient $G/{\bf O}_{3'}(G)$ isomorphic to ${\rm PSL}_2(2^p)$ for an odd prime $p$, with ${\bf O}_{3'}(G)$ nilpotent and ${\bf O}_2(G) \leq {\bf Z}(G)$. They establish that the non-abelian simple groups meeting the condition are restricted to ${\rm PSL}_2(2^p)$ (while ${\rm Sz}(2^p)$ can appear in related minimal simple contexts but not as quotients in the main theorem). The results also assert that $G$ is a Frattini cover of ${\rm PSL}_2(2^p)$ and provide existence arguments via the Doerk–Hawkes construction. Overall, the paper delivers a sharp structural description of the solvable radical and quotient, contributing to the understanding of how maximal-subgroup supersolvability controls the global group structure.
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 $3'$-group or $G/{\bf O}_{3'}(G)$ is isomorphic to ${\rm PSL}_2(2^p)$ for an odd prime $p$, where ${\bf O}_{3'}(G)$ denotes the largest normal $3'$-subgroup of $G$. Furthermore, in the latter case, ${\bf O}_{3'}(G)$ is nilpotent and ${\bf O}_2(G)\leq {\bf Z}(G)$.