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Its not showing any solution or any hint..
let me just give you the proof :
What is the case when the group becomes infinite? In the question the finite or infinite case was not mentioned...
I think we must have a "minimum positive angle of rotation" in the group for it to be cyclic; for infinite groups, we cannot guarantee the existence of a minimum rotation element in the group. so, my hunch is that there may exist some infinite groups of plane rotations that are not cyclic
for example, consider the group of rotations with angles whose measures are real numbers
then, its easy to see, since there is no minimum positive element, the previous proof using euclid theorem fails
So in general it is not cyclic, we may conclude?
On the other hand, the infinite group of rotations with angles whose measures are "integers" is cyclic. because there exists a minimum positive rotation angle : "1"
those two examples of infinite groups make me believe that some infinite groups of plane rotations are cyclic and some are not.