# Orbital maximin equals size of set for faithful actions by groups with nontrivial center

From Groupprops

## Statement

Suppose is a set of size . Consider the Orbital maximin problem (?) for faithful group actions by groups with a nontrivial center. In other words, consider the maximum possible value of the size of the smallest orbital under a faithful group action on by a group with nontrivial center. This maximum equals . (The maximum is attained by a cyclic group acting on ).

## Related facts

- Orbital maximin equals size of set for abelian groups
- Orbital maximax equals size of set for abelian groups
- Orbital maximin equals size of set for nilpotent groups
- 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