Group Theory Primer, part 2: examples of a few groups.
June 19, 2010
Last time, we talked about what a group is. This time, we’ll go over some specific groups. In the next post, we’re going to go over some basic theorems about groups.
If we take a square, say, and label the vertices A, B, C, and D, then we can do a bunch of stuff to this square. It’ll look the same if we rotate it 90 degrees, or 180 degrees, or 270 degrees — the only difference will be that the names of the vertices will have permuted around! We could even flip it around its diagonal, and the square would “look” the same, but the vertices, again, would change their letters.
In the picture above (hastily drawn in MS Paint) we see the square that I was talking about. Let’s call every rotation of the square by 90 degrees and let’s call every “flip” about the diagonal .
If the square starts with the letters like this, let’s read them off in clockwise order: they read . Good. Now, what happens if we rotate them 90 degrees? Reading from the upper left, we get . Rotating again, we get . Notice that if we rotated this square four times (or 360 degrees), we’d get back to the original picture. Okay, now that we’re back to the original picture, let’s flip this over the diagonal. We now have that A and C stay where they are, but B and D switch, so we have after a flip.
Let’s make this into a group. Let’s let be a rotation, and be the reflection, as we stated before. We notice that there is an inverse to rotating; that is, if we rotate 90 degrees, we can simply rotate 270 degrees to get back to where we started. How do we rotate 270 degrees? We simply apply three times: in other words, is the inverse of . Well, this makes sense, since ; that is, rotating 360 degrees (or, rotating 90 degrees four times) gives us the identity (which is the same as not doing anything).
What about flipping things? What if we flip twice? We get the identity back. So we have that . Alright, but what happens if we flip and then rotate? You might want to try to do this by yourself, but we have the identity: (we can follow this on the square: for the left-hand-side, we start with ABCD, then reflect to get ADCB, then rotate 90 degrees to get BADC. for the right-hand-side, we start with ABCD and rotate 270 degrees to get BCDA, and then reflect to get BADC.) — and this sums up the dihedral group. We can write the group the following way:
This is read the following way: we have two elements in our group, and , and they obey the relations on the right-hand-side. This completely characterizes the group.
This group is called the “dihedral group”, and it is confusingly denoted , which stands for “the dihedral group with 8 elements.”
Other Dihedral Groups
In general, we can take any other regular polygon with sides and make other dihedral groups in the same way. They all have the same characteristics: they have rotations that satisfy and reflections such that . We also have a slightly stranger looking relation (that weird third relation above) that is often said the following way: . If the regular polygon has sides, then the dihedral group is denoted .
There is also a , and we can define it the following way: think of the real line as our regular polygon. By “flipping”, we mean a kind of mirror reflection about the origin. By “rotating” we are going to “shift” the real line up one unit to make the origin 1 instead of 0. For example, we have that means (we apply starting at the right-side) shifting the origin up to 1 and then up to 2, then we reflect about the origin (so we flip this around 2; 1 goes to 3, 0 goes to 4, -1 goes to 5, etc.) and then we shift the origin up 1 unit, which brings us to 1 (and now our negatives are on the right-hand-side and the positives are on the left-hand-side due to the flip). Okay. Notice here that we have the same sort of deal as before: our group has and (which we make sense of by saying “we never can rotate enough to get back to where we started. this makes sense, since we’re constantly translating), and where we think of as “translating one unit left.” Think about this for a second: flip about something, translate one unit to the right, and then flip about again — what have we done? We’ve translated one unit to the left. Do it on paper if you don’t believe me! This group is one that I haven’t worked with much, but it’s useful to keep in mind as a “weird” example of a group.
The Klein-Four Group
This one is a kind of fun one. We can ordinarily create groups with “group tables.” I’m going to attempt this one below:
It’s easy to read one of these group charts, but as an example, let’s show an example with multiplication that you’ve probably seen before:
We know how to do this one: it’s just multiplication! If we want we simply go across on the top to the column that reads 3, and then down to the row that reads 2. We find that . That’s nice. Same for the chart above: if we want to find what we simply go across to column and then down to the row and find our product is, nicely enough, . Somewhat more mysterious, if we want to find the product then we go across to and down to and we find that it equals ! Surprising, but that’s the way it was defined. We can also think of this group (Called the Klein-four group and denoted, sometimes, as ) as the following
and if you notice, this is all we need to figure out any element! If we wanted, say, then we notice that . If you look up to our chart above, you’ll see that this is exactly what we wanted.
There are many others: permutation groups, matrix groups, modulo groups, and even weirder versions of groups. Generally, all we need to make groups is a set of generators (like the a’s, b’s, rotations, and reflections above) and a set of relations (like or or the like). Groups are fun, but the best way I’ve found to learn about them is to read about them and play around with them; whenever you think something is true for all groups, it’s nice to think, “oh, does this work on weird groups too?”