This is the class of primitive groups having an imprimitive subgroup of index 2. Let us make them almost simple as well. For a boring name, I will call this class (P).

So I begin with two questions for specialists:

1. Have you met this class before? In what context?
2. What do you think are the chances of giving a complete classification of them?

I have encountered these groups three times, and I want to say a bit about each of these. I am looking for connections.

The first time was probably around 40 years ago; I think I may have spoken about it in Peter Neumann’s Kinderseminar. Theorems about groups of special degrees were popular at the time. What about groups whose degree is the product of two distinct primes? At least those in class (P) can be classified (using the then-recently-announced Classification of Finite Simple Groups). To see how, the basic result is the following. By the phrase “incidence structure” I just mean a set of points and a set of lines with an incidence relation between them; automorphisms and dualities preserve incidence, and automorphisms fix (while dualities swap) the sets of points and lines. The word “lines” has no particular geometric connotation.

Theorem A primitive group in class (P) is a group of automorphisms and dualities of an incidence structure, acting on the flags (incident point-block pairs). The subgroup of index 2 consists of automorphisms.

To see this, we take the blocks of imprimitivity for the normal subgroup N to be points. Since the blocks are not fixed by G, dualities map this system to another, whose elements are the lines. The intersection of a point and a line is a block for G; since G is primitive, this intersection has at most one point. The point and line are incident if the intersection is non-empty. Now we clearly have a map from flags to the elements of the set on which G acts, and this map is a bijection preserving the action of G.

If the number of flags is pq, where p is the larger prime, then our group G has two different actions on a set of size p. By Burnside’s Theorem, this action is 2-transitive, hence (using CFSG) known,

The second encounter was in connection with the Road Closure Problem, described on this blog at https://cameroncounts.wordpress.com/2016/11/28/road-closures-and-idempotent-generated-semigroups/ . A transitive permutation group has the road closure property if, given any orbit O for G on the set of 2-subsets of its domain, and any proper block of imprimitivity B for the action of G on O, the graph with edge set O\B is connected. Groups with the RCP are precisely those such that, if t is any map on the domain whose image has size 2, the semigroup ⟨G,t⟩\G is generated by its idempotents.

Now a group with property (P) fails the road closure property, having O and B as in the definition such that there are just two translates of B in O; in other words, the stabiliser of B has index 2 in G, and is imprimitive since it preserves a disconnected graph. The converse is also true. It turns out, empirically, that most of the almost simple groups which fail the Road Closure Property have property (P), though we don’t yet have a proof.

The final encounter is described at https://cameroncounts.wordpress.com/2016/11/22/imprimitive-permutations-in-primitive-groups/ . A permutation is said to be primitive if every transitive subgroup of the symmetric group which contains it is primitive; it is imprimitive otherwise. Now, perhaps a little surprisingly, there are primitive groups made up entirely of imprimitive permutations. The complete list of these is not known. But groups with property (P) make a good starting point; for at least half of their elements (those in the normal subgroup) are imprimitive, and we only have to check the other half.