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

From Groupprops

## Statement

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

- is a Prime power (?).
- There exists a Solvable group (?) acting on a set of size with a doubly transitive group action, i.e., there is exactly one orbital. Thus, the orbital maximin is , its theoretical maximum.
- There exists a Metabelian group (?) acting on a set of size with a doubly transitive group action, i.e., there is exactly one orbital. Thus, the orbital maximin is , its theoretical maximum.
- There exists a Group satisfying Oliver's condition (?) acting on a set of size with a doubly transitive group action, i.e., there is exactly one orbital. Thus, the orbital maximin is , 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