Commutativity of the Heisenberg Group

Abstract

The Heisenberg group, named for Werner Heisenberg, is the group of upper triangular 3x3 matrices with ones down the diagonal. The group is not abelian, but shows many signs of"commutativity. It's center and commutator subgroup are as large and small (respectively) as they can be without being abelian. The nilpotent and solvable lengths are as small as they can be without the group being abelian. All of these.suggest some level.of commutativity. However when we look to the conjugacy classes and the probability of commuting we find that this commutativity dissipates, meaning it is. not as commutative as a. first look would suggest.