I'm interested in answering questions such as: find a permutation group of *n* elements restricted to the class of solvable groups for which the minimum of the sizes of orbits of subsets of size 2 is maximized. In other words, there are no "small" orbits on subsets of size two.

Here is a basic fact: if a solvable group has a 2-set-transitive action on a set, then the size of that set must be a prime power. This is because a 2-set-transitive action must be a primitive action, and so the maximal subgroup that is the stabilizer of some point has as a complement a minimal normal subgroup that turns out to be elementary abelian (because the whole group is solvable), and is in bijection with the set being acted upon. This is related to the fact that any maximal subgroup of a solvable group has prime power index.

Conversely, if *n* is a prime power, there exists a solvable group acting 2-transitively (on orbitals, i.e., ordered pairs of distinct elements, and hence also on unordered pairs). One natural construction for such a solvable group is as the affine group of a field with *n* elements (there are other constructions for some *n*).

Thus, the question is completely settled for prime powers. What can we say for other numbers?

This question occurred to me in the context of a related question that comes up when looking at a more restricted property of groups than solvable groups, called "Oliver's condition": groups that are *q*-group extensions of cyclic extensions of *p*-groups. In a joint paper (not yet published), my co-authors and I were able to show that for large enough *n*, we can use results about the distribution of prime numbers to obtain actions by groups satisfying Oliver's condition where all the orbits are of size $\Omega(n\log n)$ (unconditionally), $\omega(n^{5/4 - \epsilon})$ (conditional to the ERH) and $\omega(n^{3/2 - \epsilon})$ (conditional to Chowla's conjecture). We used this to tackle a conjecture about the decision tree complexity of graph properties. An ArXiV version of our paper (short version for conference) is available here.

Any ideas for tackling other restricted classes of solvable groups would also be much appreciated.