Lets talk about the symmetries of a square. Label the squares corners clockwise by , $2$ , $3$ , $4$ (starting from the top left). Rotating the square by $90^{\circ}$ can be represented by the function that sends

$1 \mapsto 2$

$2 \mapsto 3$

$3 \mapsto 4$

$4 \mapsto 1$ .

That is, it can be represented by the permutation $r := (1234)$ . Composing this with itself, we can get the rest of the rotations: $r^{2} = (13) (24)$ for rotating by $180^{\circ}$ , and $r^{3} = (4321)$ for rotating by $270^{\circ}$ .

We can also flip the square vertically: $f := (14) (23)$ . Notice that flipping the square horizontally is the same as flipping it vertically, and then rotating it by $180^{\circ}$ : r^2\circ f = $(13) (24) \circ (14) (23) = (12) (34)$ .

Represent the symmetries of flipping along the 1-3 diagonal, and flipping along the 2-4 diagonal in permutation notation, and as a composition of $f$ and $r$ where you flip first.

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Flipping along the 1-3 diagonal keeps 1 and 3 fixed and swaps 2 and 4, so it is equal to (24). It can be represented by $r f = (1234) (14) (23)$ . Similarly, flipping along the 2-4 diagonal is $(13) = r^{3} f$ .

Note that any symmetry has an inverse. Flipping twice in the same direction gives the trivial permutation, and rotating by $90^{\circ}$ can be reversed by rotating by $270^{\circ}$ . That is, $(24) (24) = ()$ and $(4321) (1234) = ()$ .

In total, we have eight symmetries. Rotations: $r$ , $r^{2}$ , $r^{3}$ ; flips $f$ , $r f$ , $r^{2} f$ , $r^{3} f$ ; and the trivial symmetry $e := ()$ . They are realized by permutations of the set with four elements. However, not all permutations of the set with four elements are symmetries of the square. The permutation $(12)$ would twist the top of the square while leaving the bottom fixed.

Squares aren't special though. We could have done the same thing with a pentagon, or any regular polygon. We could have done the same thing with a dodecahedron to get a subset of permutations of a 20 element set. We could go to higher dimensional polytopes. We could do the same thing with a circle to get an infinite collection of symmetries, or with an infinite ladder where moving the whole ladder up by one rung is a symmetry.

Now instead of studying these collections of symmetries case by case, it would be good to have a general theory of symmetry. Analysis that might be repeated in the above cases could be merely special cases of a general theory. This is the spirit of abstract algebra.

First, lets start with a set $G$ , which we will think of as our set of symmetries.

For any two symmetries, we can compose them to get a new one. So we should have a "composition" operation $\circ : G \times G \to G$ .