cosets of Gx. The standard quotient theorem of set theory then gives a natural bijection between and Gx. Specifically, the bijection is given by . This result is known as the orbit-stabilizer theorem. In the two cases of a small orbit, the stabilizer is non-trivial. If two elements x and y belong to the same orbit, then their stabilizer subgroups, Gx and Gy, are isomorphic. More precisely: if y = g · x, then Gy = gGx g−1. In the example this applies e.g. for 5 and 25, both reflection points. Reflection about 25 corresponds to a rotation of 10, reflection about 5, and rotation of −10. A result closely related to the orbit-stabilizer theorem is Burnside's lemma: where Xg is the set of points fixed by g. I.e., the number of orbits is equal to the average number of points fixed per group element. For the identity all 30 points are fixed, for the two rotations none, and for the three reflections two each: and Thus, the average is six, the number of orbits. Representation theory Up to isomorphism, this group has three irreducible complex unitary representations, which we will call (the trivial representation), and , where the subscript indicates the dimension. By its definition as a permutation group over the set with three elements, the group has a representation on by permuting the entries of the vector, the fundamental representation. This representation is not irreducible, as it decomposes as a direct sum of and . appears as the subspace of vectors of the form and is the representation on its orthogonal complement, which are vectors of the form . The nontrivial one-dimensional representation arises through the groups grading: The action is multiplication by the sign of the permutation of the group element. Every finite group has such a representation since it is a subgroup of a cyclic group by its regular action. Counting the square dimensions of the representations (, the order of the group), we see these must be all of the irreducible representations. A 2-dimensional irreducible linear representation yields a 1-dimensional projective representation (i.e., an action on the projective line, an embedding in the Möbius group ), as elliptic transforms. This can be represented by matrices with entries 0 and ±1 (here written as fractional linear transformations), known as the anharmonic group: order 1: order 2: order 3: and thus descends to a representation over any