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In a previous lesson, we looked at the symmetries of a triangle. This constituted the dihedral group of degree 3, which we denote $D_6$. We also generalized this, to think of the symmetries of any regular n-gon. This group is denoted $D_{2n}$.
You can think of $D_{2n}$ as permuting the vertices of the figure. A rotation permutes them by shifting them around in a clockwise motion. A reflection exchanges some of the vertices.
We can even think of other groups as likewise “permuting stuff”. For example, let’s try it with the group $\Bbb Z$ (the group of integers under addition). You can think of the operation “adding 1” as a way of permuting integers. This permutation sends 0 to 1, and 1 to 2, and 2 to 3, and so on. Likewise “adding 2” permutes the integers, sending 0 to 2, and 2 to 4, and so on.
In fact, we will eventually see that groups and permutations are nearly the same thing.
Hopefully that's some pretty good motivation for you to care about the example of permutation groups!