  In a recent paper Carot et al. considered the definition of cylindrical
symmetry as a specialisation of the case of axial symmetry. One of their
propositions states that if there is a second Killing vector, which together
with the one generating the axial symmetry, forms the basis of a
two-dimensional Lie algebra, then the two Killing vectors must commute, thus
generating an Abelian group. In this paper a similar result, valid under
considerably weaker assumptions, is derived: any two-dimensional Lie
transformation group which contains a one-dimensional subgroup whose orbits are
circles, must be Abelian. The method used to prove this result is extended to
apply three-dimensional Lie transformation groups. It is shown that the
existence of a one-dimensional subgroup with closed orbits restricts the
Bianchi type of the associated Lie algebra to be I, II, III, VII_0, VIII or IX.
Some results on n-dimensional Lie groups are also derived and applied to show
there are severe restrictions on the structure of the allowed four-dimensional
Lie transformation groups compatible with cyclic symmetry.
