Characterization of the solvable radical by Sylow multiplicities

Abstract

Let G be a finite group and let p1,...,pm be the distinct prime divisors of |G|. Given a sequence P=P1,...,Pm, where Pi is a Sylow pi-subgroup of G, and g∈G, denote by mP(g) the number of factorizations g=g1⋯gm such that gi∈Pi. Previous work (of which a significant part is in collaboration with Gil Kaplan) showed that these numbers determine the solvability of G and suggested that they may also provide a new characterization of the solvable radical. The present paper proves that the subgroup of G consisting of all g∈G such that mP(gx)=mP(x) for all P and all x∈G, is the solvable radical of G. Consequently, the solvable radical of G is the kernel of the Average Sylow Multiplicity Character, the character that assigns to each g∈G the average of mP(g) over all possible P (up to a global normalization constant).