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I don't know how to even approach this problem.

Let G be the group of rotations of a plane about a point P in the plane. Thinking of G as a Group of permutations of the plane, describe the orbit of a point Q in the plane.

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    $\begingroup$ If you apply the rotations about $P$ to a point $Q$, you get the set of all points obtained by rotating $Q$ around $P$ - i.e. you get the circle around $P$ containing $Q$. This should be obvious on some level. $\endgroup$ – anon Nov 14 '13 at 3:41
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Since rotations are isometries that preserve distance, each image of Q under this permutation is equidistant from P . Hence, the orbit of Q is a circle with center P and radius d(P, Q).

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