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.