Permutation Groups and Orbits on Power Sets

TL;DR

This paper investigates the minimal asymptotic growth rate of the number of set-orbits under permutation groups with restricted composition factors, providing bounds for groups avoiding large alternating quotients.

Contribution

It establishes the infimum of the normalized logarithm of the number of set-orbits for permutation groups with specific composition factor restrictions.

Findings

Determines the lower bound of the growth rate of set-orbits.

Provides a characterization of groups with restricted composition factors.

Advances understanding of permutation group actions on power sets.

Abstract

Let $G$ be a permutation group of degree $n$ and let $s(G)$ denote the number of set-orbits of $G$. We determine $\inf(\frac {\log_2 s(G)} n)$ over all groups $G$ that satisfy certain restrictions on composition factors (i.e. $Alt(k), k > 4$ cannot be obtained as a quotient of a subgroup of $G$).