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

✍ Scribed by Shalom Eliahou; Michel Kervaire

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 172 KB
• Volume: 124
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.09.002

✦ Synopsis

Let G be a group. We study the minimal sumset (or product set) size μ G (r, s) = min{|A • B|}, where A, B range over all subsets of G with cardinality r, s respectively. The function μ G 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 μ G (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 μ G for a few non-abelian groups of small order. These results extend the Cauchy-Davenport theorem, which determines μ G (r, s) for G a cyclic group of prime order.