Monolithic groups with all maximal subgroups of product type

Question

Dear All,

I am dealing with a problem relating to monolithic group.

Let $L$ be a monolithic group with socle $N = S^r$, where S is a nonabelian simple group. Consider the projection $p:N \to S$. A maximal subgroup $H$ of $L$ is of product type if $HN=L$ and $ 1< p( H\cap N ) < S $.

I am considering four simple groups : $PSL_3(2), PSL_4(2), PSL_2(11)$ and $PSp_4(3)$. If $S$ is one of those, then do all $L$ with socle component $S$ have property that all maximal subgroups of $L$ are of product type?

Do we have any classification of L in general? And if they do exist answers, could you please tell me some source to find them.

Thanks a lot in advance.

Answer 1

If I am understanding your question correctly, then the answer must be no. You do not say why you are interested in those four simple groups in particular, but for any finite nonabelian simple group and any such $L$ with unique minimal normal subgroup $N=S^r$ with $r>1$, there will be maximal subgroups of $L$ containing $N$, and they are not of product type.

There are other types of maximal subgroup that are not of product type. For example, for any $S$, let $L$ be the wreath product of $S$ with a cyclic group of order 2. This has maximal subgroups $H$ of diagonal type, which are the normalizers in $L$ of diagonal subgroups of $N$. These satisfy your first condition, that $HN=L$, but $p(H \cap N) = S$ for all $r$ projections $p:N \to S$, so they do not satisfy the second condition.

Answer 2

I think here you should find the answer to your question:

Monolithic primitive groups without diagonals

Answer 3

Thank Derek for your answer. I am sorry for not giving reason why I am interested in those groups. They are simple groups of Lie type over field of characteristic p that have a subgroup of p-power index (Following the paper of Gulranick on "Subgroups of prime power index in a simple group"). And I would like to know if I can (possibly) know some about some Monolithic groups having them as socle component.

Actually, I would like to know the behaviour of the number

$ a_{p^x}=\sum_{\substack{L=HN, |L:H|=p^x } }\mu_L(H)$

where $\mu_L$ is the Mobius function defined over the subgroups lattice of $L$. In fact, I would like to prove those numbers are negative.

If the characteristic of the field is different from $p$, then I know some machinary to deal with those numbers, but for the the case the characteristic is $p$, I dont know. In the key I know, if all maximal subgroups of $L$ supplementing $N$ are of product type, then it will be done. Otherwise, I dont know how to deal with it. And I just have 4 cases of simple groups as above.

Anyway, thank Derek a lot.