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.



Monolithic groups with all maximal subgroups of product type
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.


