Minimal sumsets in finite solvable groups

Given a group G and positive integers r, s <= vertical bar G vertical bar, we denote by mu(G)(r, s) the least possible size of a product set AB = {ab vertical bar a is an element of A, b is an element of B}, where A, B run over all subsets of G of size r, s, respectively. While the function mu(G) is completely known when G is abelian [S. Eliahou, M. Kervaire, Minimal sumsets in infinite abelian groups, journal of Algebra 287 (2005) 449-457], it is largely unknown for G non-abelian, in part because efficient tools for proving lower bounds on mu(G) are still lacking in that case. Our main result here is a lower bound on mu(G) for finite solvable groups, obtained by building it up from the abelian case with suitable combinatorial arguments. The result may be summarized as follows: if G is finite solvable of order m, then mu(G)(r, s) >= mu(G')(r, s), where G' is any abelian group of the same order m. Equivalently, with our knowledge of mu(G'), our formula reads mu(G)(r, s) >= min(h/m) {(inverted right perpendicularr/hinverted left perpendicular + inverted right perpendiculars/hinverted left perpendicular - 1) h}. One nice application is the full determination of the function mu(G) for the dihedral group G = D(n) and all n >= 1. Up to now, only the case where n is a prime power was known. We prove that, for all n >= 1, the group D(n) has the same mu-function as an abelian group of order vertical bar D(n)vertical bar = 2n. (C) 2009 Elsevier B.V. All rights reserved.