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

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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
nilpotent group (this page)

Related facts about orbital maximin

Type of group Result
abelian group Orbital maximin equals size of set for abelian groups
nilpotent group Orbital maximin equals size of set for nilpotent groups
group with nontrivial center, acting faithfully Orbital maximin equals size of set for faithful actions by groups with nontrivial center
solvable group 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