Group Theory 4: Symmetries
29 May 2008 – 6:37 am by Rich CochraneIn the previous instalment we saw some very “mathematical” examples of groups, but group theory also has an important geometrical aspect. We’ll take a look at that before we get into the more abstract material of group theory proper.
Rotations
“Symmetry” is a concept we learned roughly at school, and we think we can recognise when something is symmetrical and when it isn’t. In fact many things are symmetrical (or nearly so), including many things in nature.
But what is a symmetry of a shape? It’s a reversible transformation that leaves the shape looking the same as it did before. By “reversible” we mean “after the transformation has been done and then reversed, each point in the shape ends up back where it started”. Although the symmetry leaves the shape looking the same, it needn’t leave any of the points where they originally were.
To see what symmetries look like, let’s start with a simple shape like a square rather than something complicated like the natural forms above:
One transformation you can perform on this square is to rotate it by 90 degrees:
It looks the same, and what’s more this rotation can be reversed by rotating back again by 90 degrees or, alternatively, rotating all the way around again by 270 degrees. So this rotation is a symmetry of the square, and so is the rotation by 270 degrees, since that can be undone by rotating by a further 90. There’s another rotational symmetry too — by 180 degrees, which can be undone by repeating the same rotation again. Notice that none of these rotations leaves all the points where they originally were; in fact all of them move all the points except the centre of the square, which stays put.
These are all examples of what are called “rotational symmetries”, for obvious reasons, and in fact they can be turned into a group in a very natural way. Let’s start by giving them abbreviations:
| r1 | = | rotation by 90° |
| r2 | = | rotation by 180° |
| r3 | = | rotation by 270° |
These form a set, but what binary operation can we place on them to form a group? The most natural one is “do this, then do that”, which mathematicians call “function composition”. If x and y are symmetries of any kind, then xºy means “do y, then do x”. Yes, I know it’s backwards; there’s a good reason for that, but it might take us slightly too far afield to explain it.
The point is that if you apply one symmetry, you leave the square looking like the original square did (that’s the definition of a symmetry, remember). If you then apply another symmetry, it must end up looking the same again. In other words, the composite of two symmetries is again a symmetry. This is the key idea behind symmetry groups.
Cayley Tables
Let’s find out whether the set {r1, r2, r3} forms a group under composition. Now’s as good a time as any to introduce the idea of a Cayley table, which is a very simple way of visualising a group and will help us to answer this question.
The idea is to write out the set as the column-headings of the table, and also the row-headings:
r1 r2 r3 r1 r2 r3
We now fill in the table — if we can — by calculating
[row heading] º [column heading]
for each square. So for example this square:
r1 r2 r3 r1 r2 r3
repesents doing r3 and then doing r2 — that is, rotating the square by 270° and then by another 180°. What do you think the result should be? Well, 270° + 180° is 450°, and 360° is a full circle, so we can take 450° - 360°, which is 90°. This tells us that r2º
r1 r2 r3 r1 r2 r1 r3
Let’s try to fill in the rest of the table. How about this square?
r1 r2 r3 r1 r2 r1 r3
We can try calculating again: r2 means “rotate by 180°” and the square represents r2ºr2, meaning “rotate by 180° and then do it again”. What happens when you do this? Of course: two half-turns make a whole turn, and you end up back where you started. Composing r2 with itself “does nothing”.
Because of this, we don’t have closure, since “doing nothing” isn’t in our original list of three rotations. Rather than giving up, though, we look at this like a mathematician would: of course “doing nothing” is a symmetry of the square! It’s a transformation that leaves the shape looking the same as it did before, and it’s reversible (simply do nothing again; you’re already “back where you started”).
By convention, we call this symmetry “e”, and adding it, and doing a little calculation, enables us to complete the Cayley table:
e r1 r2 r3 e e r1 r2 r3 r1 r1 r2 r3 e r2 r2 r3 e r1 r3 r3 e r1 r2
Just by the fact that every entry in the table is an element of {e, r1, r2, r3} tells us that the set is closed under the binary operaton; but closure isn’t all we require for a group.
Checking the Group Axioms
We need to go back to the definition now and check all the “axioms” — the four criteria a group must satisfy — do in fact hold.
Well, an identity element is easy enough. If you do something, then do nothing, you’ve just done the original something. Likewise, if you do nothing, then do something, you just did the something you did. Sometimes algebra is clearer than English:
xºe = eºx = x
so e — the “do nothing” symmetry — is our identity element. A look at the Cayley table confirms it.
What about inverses? Remember here we’re requiring that for every element in the set there’s another element so that composing them, in either order, gives e as a result. Again looking at the Cayley table confirms that such inverses do exist; for example, the inverse of r1 is r3:
e r1 r2 r3 e e r1 r2 r3 r1 r1 r2 r3 e r2 r2 r3 e r1 r3 r3 e r1 r2
Checking all the others shows that we have an inverse for every element.
So we’ve got closure and identity and inverses — all we need now is associativity. This is much harder to show in general, so we’ll rely on a theorem we looked up in a book (there’s nothing disreputable about this) that states that composition is associative. Since we have all four properties, our set of rotations, plsu the do-nothing symmetry, form a group under composition.
Other Symmetries
There are a few more symmetries of the square too. For instance, you can “flip” the square over by reflecting it a vertical line through its centre; this is the kind of symmetry exhibited by some of the fish and insects depicted above (they don’t have rotational symmetry). There are other lines you can flip on too; the horizontal one through its centre, or either of the ones connecting diagonally opposite corners.
We can use these flips to form a number of new symmetry groups. Any one flip plus the identity element gives you one example. Another important one is formed by taking all the flips, plus the rotations, plus the identity element. This contains all the possible symmetries of the square and is called the dihedral group of the square, usually written D4. The 4 is because it’s a regular 4-sided polygon; D4 contains all the symmetries of the equilateral triangle, D5 does the job for the regular pentagon and so on.
Question: Why don’t all the flips, plus the identity element, form a group?
In some applications it may make sense to consider flips and in others it might not. Likewise, if you have an infinite, repeating pattern (like a tiling or wallpaper pattern) you might consider adding translations — moving the whole pattern by some fixed distance in a fixed direction — to the group you’re studying. Your definition of what counts as a “symmetry” depends on what sort of problem you’re trying to solve.
Taking Stock
We’ve seen examples of a number of groups:
- The groups of infinite order <Z, +>, <Q, +>, <Q*, ×> and <nZ, +>.
- The modular groups <Zn, +n> of order n.
- The dihedral groups Dn for n > 2
I’m sure you can imagine there are many other groups besides these, including the symmetry groups of shapes other than regular polygons (like the creatures in the picture at the top of this post). But so far we’ve seen the definition of a group and some examples — we haven’t really proved anything interesting about them. We’ll do a little group theory proper in the next instalment.