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

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

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.

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, $\left(24\right)\left(24\right)=\left(\right)$ and $\left(4321\right)\left(1234\right)=\left(\right)$.

In total, we have eight symmetries. Rotations: $r$, $r^2$, $r^3$; flips $f$, $rf$, $r^{2}f$, $r^{3}f$; and the trivial symmetry $e:=\left(\right)$. 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 $\left(12\right)$ 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 $\circ :G\times G\to G$.