Concrete groups (Draft)

by Daniel Satanove Oct 21 2016

Instead of thinking of a group as a set with operations satisfying axoims, we develop groups as symmetry groups of various objects

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.