Minimal sumsets in finite solvable groups

AbstractGiven a group G and positive integers r,s≤|G|, we denote by μG(r,s) the least possible size of a product set AB={ab∣a∈A,b∈B}, where A,B run over all subsets of G of size r,s, respectively. While the function μG 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 μG are still lacking in that case. Our main result here is a lower bound on μG 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 μG(r,s)≥μG′(r,s), where G′ is any abelian group of the same order m. Equivalently, with our knowledge of μG′, our formula reads μG(r,s)≥minh∣m{(⌈rh⌉+⌈sh⌉−1)h}.One nice application is the full determination of the function μG for the dihedral group G=Dn 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 Dn has the same μ-function as an abelian group of order |Dn|=2n