# Orbital maximin equals size of set for abelian groups

## Contents

## Statement

Suppose is a natural number greater than . Then, for any abelian group acting on a set of size , the maximum possible value of the minimum of the sizes of the orbitals is . Further, there exists an abelian group (namely, the cyclic group of order ) for which this maximum is achieved.

This is a resolution to the Orbital maximin problem (?) for abelian groups.

## Proof

We assume that is at least . The result holds trivially for .

### Reducing to the case of a transitive action

Suppose the action of on a set of size has more than one orbit. Let be the size of the smallest orbit, which we call . restricts to a permutation action on . In particular, the orbit of any orbital with both points in lies completely inside the orbital set for .

Thus, if we prove the result for all for a *transitive action*, then using would complete the proof. We can thus assume that the action is transitive.

### Every element must act freely (semiregularly), i.e., without fixed points

We assume acts transitively on a set of size .

Suppose we have an element in and elements such that and . Since acts transitively, there exists such that . Then, whereas . Thus, , contradicting abelianness.

Note that we can in fact deduce that the action must be a regular group action since it is both transitive and semiregular. This is part of a more general idea: any action that centralizes a transitive group action must be semiregular.

### The size of an orbital is exactly for a transitive action

As before, acts transitively on a set of size .

Consider any pair . We know that for any two elements , if , then . Since the action is free, this forces , also forcing . Thus, no two distinct elements of the orbital of can have the same first coordinate. Since the number of first coordinates is , we obtain that the size of the orbital is exactly .