On the probability of being a deficient square group on 2-element subsets

Abstract

ABSTRACTTwo elements x and y of a group G satisfy the deficient square property on 2-subsets if |{x,y}2|<4. Let ds(G) be the probability that two randomly chosen elements x and y of G satisfy the deficient square property, that is, xy = yx or . Freiman in 1981 showed that ds(G) = 1 for a finite group G if and only if G is a direct product of the quaternion group of order 8 with an elementary abelian 2-group. We show that if ds(G)<1, then , with equality if and only if G is a direct product of the dihedral group of order 8 with an elementary abelian 2-group.