Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and at least one of its maximal subgroup isn't of prime order.
Do there exist two distinct non-trivial subgroups $H_1$ and $H_2$ of $G$ such that
- $H_1$ and $H_2$ are contained in a common maximal subgroup $M$
- For each maximal subgroup $M'$ of $G$, either $H_1 \leq M'$ or $H_2 \leq M'$ (or both)?
If it is hard to answer the question in general, can we answer it for certain classes of finite groups (finite simple groups, symmetric groups,…)?
Best Answer
The alternating group $A_5$ shows that this cannot hold in general. This group has five maximal subgroups of index $5$, the point stabilizers. If the stated condition holds, then at least three of these would have to contain either $H_1$ or $H_2$. But in this group, the intersection of any three point stabilizers is trivial.