# On the subgroup lattice of a group

A couple of months ago, I went to a very enjoyable conference in Florence. During the course of this conference, I found myself stood in a coffee queue next to Peter Pal Palfy, whom I had heard of but never had the privilege of meeting. I want to recount a very interesting bit of maths that he told me about while we were in that queue – although it concerns a very natural group theory question, I had never heard of it before.

Anyone who has done a little bit of group theory is familiar with the subgroup lattice of a group. This question concerns what can happen between a *second-maximal subgroup* of a group *G*, and the group *G* itself.

A **second-maximal subgroup**, *H*, of *G* is a group that is maximal in the lattice of subgroups of *G*, once we have removed *G* itself, along with all maximal subgroups of *G*. By definition, the only proper subgroups that contain such a group *H* are maximal subgroups of *G*. So we have this set-up:

So, observe, that our second-maximal subgroup *H* lies inside precisely *n* maximal subgroups of *G*. Now, here’s the question: *what are the possibilities for the number n in the diagram?*

For instance, is it possible to find a group *G* which has a second-maximal subgroup *H* that lies in precisely 8 maximal subgroups of *G*. If the answer is “yes” to this question, then let’s say that 8 is an **sml-number**.

A preliminary observation: If *G* is elementary abelian of order *p^2* for some prime *p*, then the trivial subgroup is second-maximal, and basic linear algebra says that it lies in precisely *1+p* maximal subgroups of *G*. Thus, *p+1* is an sml-number, for every prime *p*.

With a bit of ingenuity, one can generalize this example to conclude that *q+1* is an sml-number, for every prime power *q*. What is more one can construct these examples so that the groups in question are solvable. Now, here’s a neat partial converse, due to Palfy and Pudlak: *If H is a second-maximal subgroup of a solvable group G, then H is contained in q+1 maximal subgroups for some prime power q*.

For solvable groups, then, we have a complete answer. What about non-solvable groups? Here, the situation is still unclear. A 1983 paper of Walter Feit established that both 7 and 11 are sml-number – remarkably, he does this by exhibiting two different second maximal subgroups *H* of the alternating group *A_31*, one of which lies in exactly 7 maximals (here *H* is the normalizer of a Sylow 31-subgroup in *PSL(5,2)*), the other in 11 (here *H* is the normalizer of a Sylow 31-subgroup in *PSL(3,5)*).

Subsequent work by Lucchini has shown that all numbers of the form *q+2* are sml-numbers for *q* a prime power. He has exhibited other infinite families of sml-numbers, although as far as I know Feit’s examples remain sporadic.

To my knowledge, a complete answer still remains to be proven. I find it quite remarkable that such a basic question should throw up such bizarre and sporadic behaviour, and thereby resist a complete solution…