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 k-sets if |X²| < k² for any k-element subset X of G. For each group G, there is a unique m = m_G 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 m_G = d.