Conjugacy in Groups of Upper Triangular Matrices

We show that in the groupUn( 2), a Sylow 2-subgroup ofGLn( 2), there are elements not conjugate to their inverses ifn ≥ 13. It follows that there are irreducible characters of this group that are not real-valued, and that a conjecture of Kirillov is not correct as stated. We give a partial explanation of why these group elements do not exist ifn ≤ 12, and an example to show that analogous phenomena can occur for odd primes.