Minimal sumsets in finite solvable groups

Abstract

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