  Long ago, Arad and Herzog (AH) conjectured that, in finite simple groups, the
product of two conjugacy classes of length greater than one is never a single
conjugacy class. We discuss implications of this conjecture for non-abelian
anyons in 2+1-dimensional discrete gauge theories. Thinking in this way also
suggests closely related statements about finite simple groups and their
associated discrete gauge theories. We prove these statements and provide some
physical intuition for their validity. Finally, we explain that the lack of
certain dualities in theories with non-abelian finite simple gauge groups
provides a non-trivial check of the AH conjecture.
