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

ABSTRACT

Two 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 $x^2=y^2$. 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 $ds\left(G\right)\le \frac{27}{32}$, with equality if and only if G is a direct product of the dihedral group of order 8 with an elementary abelian 2-group.

KEYWORDS:

• Commutativity degree
• deficient square

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20P05
• Secondary: 20D60