The Average Sylow Multiplicity Character and the Solvable Residual

Abstract

Let G be a finite group, and let p₁,…, p_m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P₁,…, P_m, of Sylow p_i-subgroups of G, and g ∈ G, denote by m_𝒫(g) the number of factorizations g = g₁…g_m such that g_i ∈ P_i. The properly normalized average of m_𝒫 over all 𝒫 is a complex character over G whose kernel contains the solvable radical of G [7]. The present paper characterizes the solvable residual of G in terms of this character.

Key Words:

• Characters
• Finite groups
• Solvable residual
• Sylow subgroups

2000 Mathematics Subject Classification:

• 20D20
• 20D25
• 20C15

Acknowledgments

¹All groups considered in this paper are assumed to be finite.

Notes

Communicated by A. Turull.