Is it possible to classify finite simple groups whose every maximal subgroups are not of prime order? Is it possible to answer to this question in the class of finite groups?



maximal subgroups of finite simple groups
**Prop:** If $G$ is a finite simple group, then a maximal subgroup of $G$ is trivial or has composite order


**Proof:** A maximal subgroup of $G$ being trivial clearly corresponds to $G$ being cyclic of prime order. Assume, then, that $G$ is non-abelian.


If $G$ has a maximal subgroup $C$ of prime order, then the action of $G$ on cosets of $C$ is Frobenius. Thus $G$ is a Frobenius group, $C$ is a Frobenius complement and $G$ contains a Frobenius kernel, i.e. $G$ is not simple. **QED**


So this answers your first question. As for your more general question about finite groups. Well, again, if a group has a maximal subgroup of prime order, then it is Frobenius, so you should consult the literature on Frobenius groups. For this I particularly recommend Isaac's "Finite Group Theory" and Passman's "Permutation groups".


Examples of groups with a maximal subgroup of prime order include dihedral groups of order $2m$ ($m$ odd) or, more generally $C\_n \rtimes C\_p$ where $p$ is a prime and $C\_p$ acts semi-regularly on $C\_n$. 


