Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

up vote 6 down vote accepted

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 $C_n$.

share|improve this answer
    
There can be maximal subgroups of prime order which are normal. –  Geoff Robinson Jan 3 '13 at 21:35
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.