# Orbital maximax is bounded below by constant fraction of number of ordered pairs of distinct elements for groups of fixed prime power order

## Statement

Suppose $p$ is a (fixed) prime number. Consider the Orbital maximax problem (?) for finite $p$-groups acting on a set $S$ of size $n$: we want to find the maximum possible size of the largest orbital under the action of a finite $p$-group on $S$.

The claim is that there is a constant $c_0$ (depending on $p$, and in fact of the order of $p^2$), such that for any $n$, there is a group action such that size of the largest orbital is $\ge n(n-1)/c_0$.

## Related facts

### Related facts about orbital maximax

Type of group Result
abelian group Orbital maximax equals size of set for abelian groups