MONOMIAL AND MONOLITHIC CHARACTERS OF FINITE SOLVABLE GROUPS

Let G be a finite solvable group and let p be a prime divisor of $|G|$ . We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.

The number of set-orbits of a solvable permutation group

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.