Permutation groups
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Rotation through phi
[Diagram goes here - download the original to see it.] Corresponds to two cycles: [Diagram goes here - download the original to see it. [Diagram goes here - download the original to see it.] Hence, the cyclical representation is Reflections sometimes leave some vertices unmoved. In this case, in the cyclical representation the unchanged vertices are simply omitted. It is assumed that if a vertex is not indicated as part of a cycle that it is not moved. For example: [Diagram goes here - download the original to see it.] This reflection cyclically permutes the vertices 2 and 4 leaving vertices 1 and 3 unchanged, and hence has cyclical representation (2 4) This discussion of the symmetries of the square in the context of permutation group S4 makes it clear that the group of all symmetries of the square is a subgroup of the permutation group S4. For example - the permutation (1 4) is not a symmetry of the square [Diagram goes here - download the original to see it.] It would involve twisting the square in a way that is not acceptable as a symmetry. There are 4! = 24 permutations of 4 symbols, but there are 8 symmetries of a square. [Diagram goes here - download the original to see it.] The symmetries of a square comprise four reflections and four rotations. The four rotations include the identity symmetry.
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Contents of
Permutation groups
1 Permutation groups
2 Symmetry groups are also permutation groups
3 Symmetry group of the square
4 Rotation through phi
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