Permutation Groups and Set-Orbits on the Power Set

A permutation group G acting on a set Ω induces a permutation group on the power set P ( Ω ) (the set of all subsets of Ω ). Let G be a finite permutation group of degree n and s ( G ) denote the number of orbits of G on P ( Ω ) . It is an interesting problem to determine the lower bound inf log 2 s ( G ) n over all groups G that do not contain any alternating group A ℓ (where ℓ > t for some fixed t ⩾ 4 ) as a composition factor. The second author obtained the answer for the case t = 4 in Yang (J Algebra Appl 19:2150005, 2020). In this paper, we continue this investigation and study the cases when t ⩾ 5 , and give the explicit lower bounds inf log 2 s ( G ) n for each positive integer 5 ⩽ t ⩽ 166 .