Conjugacy in Groups of Upper Triangular Matrices

Abstract

We show that in the groupUn(F2), a Sylow 2-subgroup ofGLn(F2), 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. © 1998 Academic Press.