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Handout Wednesday 9/24
Let’s look at a few more examples of groups. Last time we considered symmetries of a square. What about symmetries of a triangle? Instead of using a physical triangle to investigate this, let’s consider just what happens to the corners and develop some notation for tracking this.
While the symmetries of a triangle correspond to rigid motions of the plane, each corresponds exactly to simply a permutation of the vertices of that triangle. So we can consider the different permutations of three objects, call them \(1\text{,}\) \(2\text{,}\) and \(3\text{.}\)
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Use two-line notation for this.
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It might be easier to use cycle notation though. For example, the permutation that swaps 1 and 2 and leaves 3 fixed can be written as \((12)\text{.}\) The cycle \((132)\) means that 1 goes to position 3, 3 goes to position 2, and 2 goes to position 1.
