The Average Sylow Multiplicity Character and Solvability of Finite Groups

Abstract

Let G be a finite group, and let p ₁,…, p _m be the distinct prime divisors of |G|. A complete Sylow product of G is a product P ₁,…, P _m , where each P _i is a Sylow p _i -subgroup of G, 1 ≤ i ≤ m. Let R(G) be the solvable radical of G, and let H(G) be the intersection of all complete Sylow products of G. It is known ([8]) that R(G) ≤ H(G) for any G, while characterizing the groups G for which R(G) = H(G) is an open question. New results (Corollary 15) about this question are obtained by studying the average Sylow multiplicity of PSL(2, q). The multiplicity of g ∈ G in a sequence P ₁,…, P _m of Sylow p _i -subgroups is the number of factorizations g = g ₁…g _m such that g _i  ∈ P _i . It is proved that the properly normalized average of the multiplicity over all Sylow sequences as above (for a fixed ordering of the primes), is a character of G, and a general formula for its irreducible decomposition is given. It is shown that G is solvable iff the average Sylow multiplicity character is the trivial character, and the connection to a solvability criterion of Gallagher is explained.

Key Words:

• Solvable radical characters
• Sylow sequence

2000 Mathematics Subject Classification:

• 20D25
• 20C15

ACKNOWLEDGMENTS

I would like to thank Marcel Herzog for his kind hospitality, keen interest, and good advice. I would also like to thank Gil Kaplan, my friend and colleague, for his constant encouragement.

Notes

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

²The trivial group is defined to be unique1.

Communicated by A. Turull.