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Conjugacy in Groups of Upper Triangular Matrices

Abstract We show that in the group U n ( F 2 ), a Sylow 2-subgroup of GL n ( F 2 ), there are elements not conjugate to their inverses if n  ≥ 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 if n  ≤ 12, and an example to show that analogous phenomena can occur for odd primes.