 Okay so I want to take one more lesson just to look at some more interesting groups. Yesterday we looked at the integers mod n and today I'm going to start with a very simple group. My group A is going to be the fourth root of one. In other words it's the fourth roots of one and what are they? Well that is just a set of four elements one minus one the imaginary number i and negative the imaginary number i. So take any one of those and multiply it by itself four times one times one times one times one is one negative one times negative one times negative one times negative one that's one i times i is negative one and that squared is again one and negative i to the power four that's also going to be one and the binary operation is going to be multiplication. Multiplication so let's see if we have a group here consisting of A under multiplication. Is that a group or not? So let's just very quickly go to a Cayley's table we have multiplication we have one we have negative one we have i and we have negative i so on this side we must have one negative one we must have i and negative i there we go see if you can do this one times one is one negative one i negative i and same here one i negative i so it seems as we already have an identity element let's just make sure and carry on with that negative one times negative one is one negative one times i is negative i and negative one times negative i that's this going to be i i times negative one is negative i i times i is negative one and i times i is negative one with that negative gives me a one negative i times one is negative i that gives me an i i times i is negative one and with this negative just gives me a one and i negative and the negatives are positive and i times i is negative one so we have negative one there I hope I didn't make a tiny little error in my arithmetic there. So there is a Cayley's table. We seem to have an identity element. Our identity element of those is one. And let's see if we have unique, if we have, there's also closure under multiplication because the multiplication, any of the two of those, this leaves me with one of the elements. So we have that. I'm going to leave the associativity property to you. You'll definitely see that that holds. But let's just look at inverse. What is the inverse? What is the inverse of each of these? Of one is, so what times one in this group will give me back the identity element? So one is its own inverse. Negative one, what is its inverse? What times negative one gives me one again? Well, it's its own inverse i. What must I multiply i by to get one? That's negative i. And negative i, its inverse is, negative i's inverse is i. So we see we have all of them there, one negative one, negative i and i. So they, each one is in there and each one's inverse in the other way around is unique. So we do have that. The fourth roots of one under the binary operation of multiplication is indeed a group. I'm going to clear the board. Let's look at one that's slightly more difficult. Here's our second example, which is going to be slightly more difficult. It is the third roots of one and we see there negative a half plus a half square root of three i, negative half minus a half square root of three i and one. And my binary operation is multiplication. So just take any one of these to the power three and of course you're going to get one. Just to make things easier, we'll call this omega one, we'll call this omega two and one, we'll call omega three. And under multiplication, here's my Cayley's table. Do this, it's nice mental exercises to do these multiplications and we see this Cayley's table. Omega one times omega one gives me omega two and what we can very quickly see here is of course we have closure. I leave you with associative property but I tell you now that it does hold and then let's just see if there's an inverse. What is the inverse? You see if those exist and are unique. Well e just seems to be, my identity element is of course omega three, which you know we can think is going to be one. Anything times one is going to be anything. So this is like the inverse of omega one. So what times omega one gives me omega three? So its inverse is, the inverse of omega one is omega two. The inverse of omega two is so two times what gives me omega three, omega two times omega one. So omega one and omega three, it's inverse. The inverse of omega three is of course omega three because if I multiply omega three by omega three, I still get omega three and we see that these are unique as well. So my inverses, they are all in that set and each of them is unique. So we see that indeed G is a group, A is a group with my binary addition of my binary operation of multiplication. So those are fun little groups and you see that groups actually they pop up everywhere.