The Average Sylow Multiplicity Character and Solvability of Finite Groups

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 Kaplan , G. , Levy , D. ( 2005 ). Sylow products and the solvable radical . Arch. Math. 85 ( 4 ): 304 – 312 .[Crossref], [Web of Science ®] , [Google Scholar]]) 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.