# Orbital maximin problem

From Groupprops

## Statement

For a given natural number and a group property , the **orbital maximin problem** for the pair is the problem of finding a group satisfying with an action on a set of size such that the minimum of the sizes of the orbitals under the action of is as large as possible.

Here, an orbital is an orbit under the induced action of on unordered pairs of distinct elements from the set.

In general, we take to be a property that is both subgroup-closed and quotient-closed. For such , we can restrict attention to faithful group actions.

This is related to the orbital maximax problem, where we try to maximize the size of the largest orbital.

## Particular cases

- Orbital maximin equals size of set for abelian groups
- Orbital maximin equals size of set for groups with nontrivial center
- Orbital maximin equals number of ordered pairs of distinct elements for solvable groups iff size is prime power
- Orbital maximin is bounded below by constant fraction of number of ordered pairs of distinct elements for solvable groups