Coherent permutation groups

This post, in a sense, follows on from the previous one, though I won’t explain the connection just yet.

Here is another property of permutation groups which occurred to me last night. I have no idea yet whether it is worth anything at all, but I am open to ideas.

Let G be a group with a normal subgroup N. We say that the extension splits if N is complemented in G, that is, there exists a subgroup H of G such that NH = G and N∩H = {1}. If this happens, then G/N is isomorphic to H.

So here is the new definition. Let G be a permutation group on a set Ω. I will say that G is coherent if, for any subset Δ of Ω, the setwise stabiliser of Δ splits over the pointwise stabiliser.

Not every permutation group is coherent. Let G be a group which does not split over a normal subgroup N, and let Ω be the disjoint union of the groups G and G/N, with the action of G by right multiplication. Let Δ be the coset space of G/N. Then the setwise stabiliser of Δ is G while the pointwise stabiliser is N. The smallest example is the cyclic group of order 4, acting with two orbits of sizes 4 and 2.

But some of our favourite examples are coherent. Examples include regular and Frobenius groups, the symmetric and alternating groups. (Coherence for the alternating groups takes a little thought; the others are clear.)

So what would I like to do?

First and foremost, show that all groups in some class are coherent, or give a classification of those that are not. This might be possible for transitive groups, but is maybe more likely for primitive groups.

And, in parallel with this, connect the property of coherence to other notions in permutation groups or elsewhere in mathematics.

Suggestions welcome!