How commutative can a non-commutative group be?

Abstract

Let G denote a finite non-commutative group. What can we say about the number of ordered pairs (x,y), where x,y ϵ G, such that xy = yx?In order to prepare ourselves to answer this question, let us examine the Cayley table of S3, the symmetric group on three symbols. We take as generators the reflection (12) and the rotation (123). It is clear that we are examining the Cayley table for symmetry about its main diagonal.