Exercise 1) Can you think of some other groups? Remember, you need to specify the set, the function, the special elements, and the inverses, and then make sure that all of the axioms in the definition of a group hold (i.e., check that the function is “order insensitive”, and so on).
Solution: In this exercise, we’ll really start to see the power and flexibility behind the definition of a group. Namely, we’ll see that a group can be used to do much more than simply describe how numbers add (or multiply) as we saw in the lesson. This will also begin to show the creativity that goes into mathematics. We’ve already had to get creative in finding novel ways to prove things, but the creativity involved in making “correct” definitions and finding interesting examples is something somewhat different (I use scare quotes on “correct” because there are no “correct” or “incorrect” definitions, just more and less interesting ones).
Check out the following example of a group. Consider taking a perfectly square piece of paper and lying it on your desk. Now consider the following four actions I can take on that square that will leave the square “looking the same” on your desk (assuming there are no marks or anything on the paper). I can 1) leave it alone (do nothing to it), I can 2) rotate it clockwise by 90 degrees, I can 3) rotate it clockwise by 180 degrees, and I can 4) rotate it clockwise by 270 degrees. Note that if I rotate it 360 degrees, I can simply say that I’ve done nothing to it, because it’s exactly where it started. Thus, we identify a 360 degree rotation with action #1 (doing nothing), and we identify a 450 degree rotation with action #2 (rotating by 90 degrees). Now, I can consider the set of these four actions. I.e., there are four elements in my set, and each element is an action on the square. My special map is “composition of actions”, meaning I take two actions (analogous to taking two integers) and define a new action (analogous to the sum of the two integers) by composing the actions one after the other (analogous to adding the integers). Let me (foreshadowing-ly) call action #1 “0”, action #2 “90”, action #3 “180”, and action #4 “270”. Then my set (call it ) is Since my special function is “composition of actions”, we have that, for example, because combining a 90 degree rotation with another 90 degree rotation yields a 180 degree rotation. Similarly, , and because two 180 degree rotations, done one after the other, is a 360 degree rotation, which is the same as “doing nothing”. Clearly, the special element is “0” (because composing an action with the action of “doing nothing” just gives you back the original action), and clearly this map is associative because it doesn’t matter if we first rotate by, say, 90, then by 180, and then by 270, or first by 180, then by 270, and then by 90—we’ll always land in the same place. Lastly, all elements have an inverse, where the inverse is just rotating “through” to 360. Thus, the inverse of 90 is 270, the inverse of 180 is 180 (180 is its own inverse!), and the inverse of 270 is 90 (the inverse of 0 is, well, 0!). So this forms a perfectly good group, and it has very little to do with numbers! Now, can you generalize this to other regular polygons? Hint: consider a stop sign. What kind of group can you define with that?
Here’s another cool example of a group. Consider the set and let’s make it so that we can add these numbers together. To make this a group, however, we have to deal with which, in “normal” addition, would give 5. But 5 isn’t in this set, and so we can’t use “normal” addition to make this set a group. We can, however, use “clock” addition (which we’ll see more of in coming lessons). We could make it so that we go “back around” the set, so that but now 1+3=0! That’s fine—this is a definition of the map that we’re using to make this set a group (i.e., we’re defining the function to be such that ). Addition will work the exact same way on the other elements, where for example but and This is indeed a group, as addition is still associative and the special element is 0. The only difference is in the inverses—“-1” isn’t in the set, but 3 is the inverse of 1 now, since 1+3=0! What are the inverses of the other elements? Can you make another group in a similar way out of the set ? (Hint: yes).
Groups arise all over mathematics, and they come in all different shapes, sizes, and forms. The more you think about groups and try to come up with cool examples, the easier the coming lessons will be and the more “mathematical sophistication” you’ll develop. We will be dealing with groups for quite some time, and once we get extremely comfortable with them, we’ll move on to even better things! Groups will, however, always be in the background.
On to Lesson 23