Products of nonconjugate maximal subgroups

Abstract

We prove that a finite group $G$ is solvable whenever $MN=G$ for every pair of nonconjugate maximal subgroups $M,N<G$. Equivalently, every finite nonsolvable group has two nonconjugate maximal subgroups whose setwise product is proper. This gives a negative answer to Problem 10.34 of the Kourovka Notebook. As an application, we also answer an open question raised by Guo.