The number of set-orbits of a solvable permutation group

Abstract

Abstract A permutation group G acting on a set Ω induces a permutation action on the power set 𝒫 ⁢ ( Ω ) {\mathscr{P}(\Omega)} (the set of all subsets of Ω). Let G be a finite permutation group of degree n, and let s ⁢ ( G ) {s(G)} denote the number of orbits of G on 𝒫 ⁢ ( Ω ) {\mathscr{P}(\Omega)} . In this paper, we give the explicit lower bound of log 2 ⁡ s ⁢ ( G ) / log 2 ⁡ | G | {\log_{2}s(G)/{\log_{2}\lvert G\rvert}} over all solvable groups G. As applications, we first give an explicit bound of a result of Keller for estimating the number of conjugacy classes, and then we combine it with the McKay conjecture to estimate the number of p ′ {p^{\prime}} -degree irreducible representations of a solvable group.