On Theta Pairs for a Maximal Subgroup

Abstract

For a maximal subgroup $M$ of a finite group $G$, a $\Theta$-pair is any pair of subgroups $(C,D)$ of $G$ such that (i) $D \triangleleft G,D \subset C$, (ii) $\left \langle {M,C} \right \rangle = G,\left \langle {M,D} \right \rangle = M$ and (iii) $C/D$ has no proper normal subgroup of $G/D$. A natural partial ordering is defined on the family of $\Theta$-pairs. We obtain several results on the maximal $\Theta$-pairs which imply $G$ to be solvable, supersolvable, and nilpotent.