A note on roots of Mobius transformations in discrete groups

This short article treats a very naive question concerning roots of Mobius transformations in discrete groups. We exhibit, according to the standard classification of Mobius transformations, formulations depending on the types of given transformations. Along the way, algebraic natures of relevant subgroups of a discrete group under consideration are discussed, which allows us to provide intuitive geometrical interpretations. We indicate how the discreteness plays a substantial role during the study while exploring folklore or classical results for these subgroups. We conclude by remarking on the original motivation for the study and point out possible directions for future research.