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

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Statement

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

  1. n is a Prime power (?).
  2. There exists a 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 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 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

Facts used

  1. Primitive solvable group acts on a set iff the set has prime power size