  For a mixing shift of finite type, the associated automorphism group has a
rich algebraic structure, and yet we have few criteria to distinguish when two
such groups are isomorphic. We introduce a stabilization of the automorphism
group, study its algebraic properties, and use them to distinguish many of the
stabilized automorphism groups. We also show that for a full shift, the
subgroup of the stabilized automorphism group generated by elements of finite
order is simple, and that the stabilized automorphism group is an extension of
a free abelian group of finite rank by this simple group.
