Commuting Pairs in Finite Nonabelian Groups

The study of the probability that two group elements commute dates back to 1968 with the work of Paul Erdos and Paul Turan. Since then, much has been deduced about these probabilities, including its bound of $5/8$ for nonabelian groups. During this talk, we will look at the associated probabilities of finite nonabelian groups and how to calculate such probabilities using several methods. Furthermore, reports will be made on known probabilities associated with dihedral groups and how to calculate probabilities with specified denominators as well as specified numerators. Finally, we will wrap up with looking at the group of $GL(2,\mathbb{Z}_p)$ matrices and deducing the probability that two of these matrices commute.