Concrete groups (Draft)

Lets talk about the symmetries of a square. Label the squares corners clockwise by $1$, $2$, $3$, $4$ (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: $f:= (1 4)(2 3)$. Notice that flipping the square horizontally is the same as flipping it vertically, and then rotating it by $180^\circ$: r^2\circ f = $(13)(24)\circ(14)(23) = (1 2)(3 4)$.

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.

%%hidden(Show solution): 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 $rf = (1234)(14)(23)$. Similarly, flipping along the 2-4 diagonal is $(13) = r^3f$. %%

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 $f$, $rf$, $r^2f$, $r^3f$; 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 $\circ : G \times G \to G$.