Orbital maximin equals number of ordered pairs of distinct elements for solvable groups iff size is prime power

Statement
Let $$n$$ be a natural number. Then, the following are equivalent:


 * 1) $$n$$ is a fact about::prime power.
 * 2) There exists a fact about::solvable group acting on a set of size $$n$$ with a doubly transitive group action, i.e., there is exactly one orbital. Thus, the orbital maximin is $$n(n-1)$$, its theoretical maximum.
 * 3) There exists a fact about::metabelian group acting on a set of size $$n$$ with a doubly transitive group action, i.e., there is exactly one orbital. Thus, the orbital maximin is $$n(n-1)$$, its theoretical maximum.
 * 4) There exists a fact about::group satisfying Oliver's condition acting on a set of size $$n$$ with a doubly transitive group action, i.e., there is exactly one orbital. Thus, the orbital maximin is $$n(n-1)$$, its theoretical maximum.

Related facts

 * Orbital maximin equals size of set for abelian groups
 * Orbital maximin is bounded below by constant fraction of number of ordered pairs of distinct elements for solvable groups

Facts used

 * 1) uses::Primitive solvable group acts on a set iff the set has prime power size