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

摘要

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.