Supersolvable Subgroups of Order Divisible by 3

Source abstract

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 $$3'$$ 3 ′ -group or $$G/\textbf{O}_{3'}(G)$$ G / O 3 ′ ( G ) is isomorphic to $$\textrm{PSL}_2(2^p)$$ PSL 2 ( 2 p ) for an odd prime p , where $$\textbf{O}_{3'}(G)$$ O 3 ′ ( G ) denotes the largest normal $$3'$$ 3 ′ -subgroup of G . Furthermore, in the latter case, $$\textbf{O}_{3'}(G)$$ O 3 ′ ( G ) is nilpotent and $$\textbf{O}_2(G)\le \textbf{Z}(G)$$ O 2 ( G ) ≤ Z ( G ) .