On finite groups and the small square property

Let G be a finite group written multiplicatively and k a positive integer. If X is a non-empty subset of G, write X2 = {xy | x, y ∈ X}. We say that G has the small square property on fc-sets if \X2\2 for any k-element subset X of G. For each group G, there is a unique m = mG such that G has the small square property on (m + 1)-sets but not on m-sets. In this paper we show that given any positive integer d, there is a finite group G with mG = d. © Akadémiai Kiadó,.