Minimal sumsets in finite solvable groups

Given a group GG and positive integers r,s≤|G|r,s≤|G|, we denote by μG(r,s)μG(r,s) the least possible size of a product set AB={ab∣a∈A,b∈B}AB={ab∣a∈A,b∈B}, where A,BA,B run over all subsets of GG of size r,sr,s, respectively. While the function μGμG is completely known when GG is abelian [S. Eliahou, M. Kervaire, Minimal sumsets in infinite abelian groups, Journal of Algebra 287 (2005) 449–457], it is largely unknown for GG non-abelian, in part because efficient tools for proving lower bounds on μGμG are still lacking in that case. Our main result here is a lower bound on μGμG 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 GG is finite solvable of order mm, then μG(r,s)≥μG′(r,s)μG(r,s)≥μG′(r,s), where G′G′ is any abelian group of the same order mm. Equivalently, with our knowledge of μG′μG′, our formula reads μG(r,s)≥minh∣m{(⌈rh⌉+⌈sh⌉−1)h}.One nice application is the full determination of the function μGμG for the dihedral group G=DnG=Dn and all n≥1n≥1. Up to now, only the case where nn is a prime power was known. We prove that, for all n≥1n≥1, the group DnDn has the same μμ-function as an abelian group of order |Dn|=2n|Dn|=2n.