Commutativity in non-Abelian Groups

Let P2(G) be defined as the probability that any two elements selected at random from the group G, commute with one another. If G is an Abelian group, P2(G) = 1, so our interest lies in the properties of the commutativity of nonAbelian groups. Particular results include that the maximum commutativity of a non-Abelian group is 5/8, and this degree of commutativity only occurs when the order of the center of the group is equal to one fourth the order of the group. Explicit examples will be provided of arbitrarily large non-Abelian groups that exhibit this maximum commutativity, along with a proof that there are no 5/8 commutative groups of order 4 mod 8. Further, we prove that no group exhibits commutativity 0, there exist examples of groups whose commutativity is arbitrarily close to 0. Then, we show that for every positive integer n there exists a group G such that P2(G) = 1/n. Finally we prove that the commutativity of a factor group G/N of a group G is always greater than or equal to the commutativity of G. 0 Introduction: The way we define Abelian groups (see Definitions 0.2 and 0.3) provides us with a simple way of understanding what might be termed the commutativity of such groups. In particular, as each pair of elements in an Abelian group necessarily commutes, we can say that the group has complete commutativity, or 100% commutativity, or, on a decimal scale of zero to one, commutativity 1. As is known by those with even a little background in the study of group theory, however, Abelian groups account for only a small proportion of all groups.