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

## Statement

The following are equivalent for a natural number $n$:

1. $n$ is a Prime power (?).
2. There exists a solvable group $G$ with a Primitive group action (?) on a set of size $n$.
3. There exists a solvable group $G$ with a Doubly set-transitive group action (?) on a set of size $n$.
4. There exists a solvable group $G$ with a Doubly transitive group action (?) on a set of size $n$.

(Note that since solvability is quotient-closed, assuming that the action in any of the above cases is faithful does not alter the strength of the statement).

## Facts used

1. Doubly transitive implies doubly set-transitive
2. Doubly set-transitive implies primitive
3. Primitive implies innately transitive: In a primitive group, there is a transitive minimal normal subgroup.
4. Minimal normal implies elementary abelian in finite solvable
5. Fundamental theorem of group actions

## Proof

### (1) implies (4)

This follows from an explicit construction: consider the field $F$ of order $n$, and consider the general affine group $GA(1,F)$, acting on the set $F$. This is doubly transitive.

### (4) implies (3) implies (2)

This follows from facts (1) and (2).

### (2) implies (1)

As remarked earlier, we can assume that the action is faithful, and hence, $G$ is a primitive group. Also, since the set is finite, $G$ is finite.

By facts (3) and (4), $G$ has a transitive minimal normal subgroup that is also elementary abelian. Thus, there is an elementary abelian group acting transitively on the set of size $n$. By fact (5), $n$ must divide the order of the elementary abelian group. Since elementary abelian groups have prime power order, $n$ must be a prime power.