The Kernel of the Average Sylow Multiplicity Character and the Solvable Radical

Abstract

Let G be a finite group, and let p 1,…, p m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P 1,…, P m , where P i is a Sylow p i -subgroup of G, and g ∈ G, denote by m 𝒫(g) the number of factorizations g = g 1…g m such that g i ∈ P i . Previously, it was shown that the properly normalized average of m 𝒫 over all 𝒫 is a complex character over G termed the Average Sylow Multiplicity Character. The present article identifies the kernel of this character as the subgroup of G consisting of all g ∈ G such that m 𝒫(gx) = m 𝒫(x) for all 𝒫 and all x ∈ G. This result implies a close connection between the kernel and the solvable radical of G.