Maximal subgroups of some special cases of ${\rm L}_{n}^{\epsilon}(q)$

Question

Let $G$ be a finite simple group of type ${\rm L}_{n}^{\epsilon}(q)$ with the following conditions:
$n$ and $\dfrac{q^{n}-\epsilon}{(q-\epsilon)(n,q-\epsilon)}$ both prime, $n\geqslant3$ and $(n,q,\epsilon)\neq(3,4,+),(3,3,-),(3,5,-),(5,2,-)$

Show that:
1- Every minimal subgroup $L$ of order $s=\dfrac{q^{n}-\epsilon}{(q-\epsilon)(n,q-\epsilon)}$ is contained properly in a maximal subgroup $M$ isomorphic to $\mathbb{Z}_{s}\rtimes\mathbb{Z}_{n}$
2- The maximal subgroup $M$ is the only proper subgroup of $G$ which contains properly $L$, i.e. every minimal subgroup of order $s$ has a unique overgroup.

As far as I checked the "ATLAS" the above propositions both are true.

Answer 1

It looks to me as though the answer is yes. You have excluded those cases where $C_s \rtimes C_n$ is not maximal in $L_n^\epsilon(q)$.

So in any other situation, $C_s \rtimes C_n$ is maximal, and hence if $C_s$ were contained in some other maximal subgroup $M$ then we would have $M \cap C_s = C_s$. But then $C_s$ would be a self-normalizing Sylow $s$-subgroup of $M$, and so $M$ would have a normal $s$-complement by Burnside's Transfer Theorem. This rules out $M$ being almost simple, and the fact that $C_s$ acts irreducibly implies that $M$ cannot be a geometric-type maximal.