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I was thinking of symmetries today and this kept bugging me. I wonder if anyone can help me with this.

Consider the symmetries of a triangle. You can do 3 rotations and 3 reflections to get them all. Number the vertices, create an ordered set with 3 elements that contain the numbers of the vertices. Starting with the topmost and working clockwise you'd get (1,2,3), (2,3,1), etc. By doing all the symmetries you can get all possible combinations of three numbers.

Consider the symmetries of a square. You can do 4 rotations and 4 reflections to get them all. Do the same procedure as above, that is, number the vertices and construct the ordered sets. Now you don't end up with all possible combinations of 4 numbers. For example: number the top left hand side vertex 1, the top right hand side 2, the bottom right 3 and the bottom left 4. I cannot think of a way of getting the set (1,2,4,3). Can anyone?

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# Symmetry of the square

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