Conjugacy classes of centralizers in the group of upper triangular matrices

Abstract

Let 𝐺 be a group. Two elements 𝑥,𝑦∈𝐺 are said to be in the same 𝑧-class if their centralizers in 𝐺 are conjugate within 𝐺. In this paper, we prove that the number of 𝑧-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least 6, and finite otherwise.