Some results on minimal sumset sizes in finite non-abelian groups

Let G be a group. We study the minimal sumset (or product set) size μG(r,s)=min{|A⋅B|}, where A,B range over all subsets of G with cardinality r,s respectively. The function μG has recently been fully determined in S. Eliahou, M. Kervaire, A. Plagne, Optimally small sumsets in finite abelian groups, J. Number Theory 101 (2003) 338-348; S. Eliahou, M. Kervaire, Minimal sumsets in infinite abelian groups, J. Algebra 287 (2005) 449-457] for G abelian. Here we focus on the largely open case where G is finite non-abelian. We obtain results on μG(r,s) in certain ranges for r and s, for instance when r?3 or when r+s?|G|−1, and under some more technical conditions. (See Theorem 4.4.) We also compute μG for a few non-abelian groups of small order. These results extend the Cauchy-Davenport theorem, which determines μG(r,s) for G a cyclic group of prime order.