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 ([Citation8]) 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.
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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.