Upper bounds on the number of conjugacy classes in unitriangular groups

Andrew Soffer

Published 2014-11-19Version 1

We provide a new upper bound on the number of conjugacy classes in the group $U_n(q)$ of unitriangular matrices over a finite field. Higman originally introduced this problem and gave the first upper bound as a means of bounding the number of $p$-groups of a given order, though the question itself is of independent interest. Our tools also allow us to compute an upper bound for every term in the lower central series of $U_n(q)$.