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

• Published in: Journal of number theory 2007-05, Vol.124 (1), p.234-247
• Main Authors: Eliahou, Shalom, Kervaire, Michel
• Format: Article
• Language: English
• Subjects: Additive Number Theory; Cauchy–Davenport theorem; Non-abelian groups; Sumsets
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/j.jnt.2006.09.002

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.