A group-invariant version of Lehmer's conjecture on heights

We state and prove an equivariant version of Lehmer’s conjecture on heights, generalizing papers by Zagier (1993) and Dresden (1998) which are special cases of this theorem. We also extend their three cases to a full classification of all groups satisfying the condition that the set of all orbits for which every non-zero element lies on the unit circle is finite and non-empty.