The symmetries of periodic solutions obtained from Hopf bifurcation in systems with finite abelian symmetries are investigated. Main result is the abelian Hopf
theorem which gives necessary and sufficient conditions for when these
periodic solutions can occur by Hopf bifurcation when
is a finite Abelian group. Examples of these results in the case when the symmetry group .
by permutation of coordinates are given. In this case the
periodic solutions that are obtainable by a generic Hopf bifurcation are classified and it is shown that there exist families of
periodic solutions that cannot be obtained by Hopf bifurcation.