  A celebrated result of J. Thompson says that if a finite group $G$ has a
fixed-point-free automorphism of prime order, then $G$ is nilpotent. The main
purpose of this note is to extend this result to finite inverse semigroups. An
earlier related result of B. H. Neumann says that a uniquely 2-divisible group
with a fixed-point-free automorphism of order 2 is abelian. We similarly extend
this result to uniquely 2-divisible inverse semigroups.
