  This is a study of fusion ring automorphisms leaving only the trivial element
fixed. We prove that a variety of classical results on fixed-point-free
automorphisms of finite groups are true in the generality of fusion rings. As a
result, we show there are $8$ Grothendieck equivalence classes of fusion
categories of rank less than $9$ with a fixed-point-free fusion automorphism of
prime order, generalizing existing results about modular fusion categories of
odd dimension.
