  We study the stabilized automorphism group of minimal and, more generally,
certain transitive dynamical systems. Our approach involves developing new
algebraic tools to extract information about the rational eigenvalues of these
systems from their stabilized automorphism groups. In particular, we prove that
if two minimal system have isomorphic stabilized automorphism groups and each
has at least one non-trivial rational eigenvalue, then the systems have the
same rational eigenvalues. Using these tools, we also extend Schmieding's
result on the recovery of entropy from the stabilized automorphism group to
include irreducible shifts of finite type.
