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

## Contents

## Statement

The following are equivalent for a natural number :

- is a Prime power (?).
- There exists a solvable group with a Primitive group action (?) on a set of size .
- There exists a solvable group with a Doubly set-transitive group action (?) on a set of size .
- There exists a solvable group with a Doubly transitive group action (?) on a set of size .

(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).

## Related facts

- Primitive implies Fitting-free or elementary abelian Fitting subgroup
- Classification of solvable transitive subgroups of symmetric group of prime degree
- Primitive solvable group acts on a set iff the set has prime power size

## Facts used

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

## Proof

### (1) implies (4)

This follows from an explicit construction: consider the field of order , and consider the general affine group , acting on the set . 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, is a primitive group. Also, since the set is finite, is finite.

By facts (3) and (4), 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 . By fact (5), must divide the order of the elementary abelian group. Since elementary abelian groups have prime power order, must be a prime power.