 Welcome back to our lecture series Math 42-20, Abstract Literature of 1 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. We're still going to be talking about groups with some definitions and examples. But in lecture 9 here, we're going to focus on group properties. We're going to prove, we'll introduce some definitions of notation, but we're actually going to prove some properties that are universally true for every single group. That is, the things we're going to prove right now are not about individual groups, but about every single group ever conceived, because what we will prove will be properties derived from the three axioms of groups, associativity, identity, and inverses. So before we get to that, let me introduce some vocabulary here. When we talk about a group, we often talk about the order of the group, right? Because when you talk about a group, there's actually two things involved. There's a set, and then there's the binary operation on that set. Now, because groups have, you know, it's a set and operation, it gets a little bit tedious to list them all the time. And oftentimes when one describes a group, you just describe the set. You're like, oh, you say G is a group, where then the operation is somewhat implied by the context. And if there were multiple operations in play and there's any ambiguity, the author would be, it would be duty called to specify the operation. But oftentimes you'll often see things like G is a group without the operation mentioned because it's implied. When one talks about the order of a group, we are talking about the cardinality of the underlying set, how big is the group. And so one important dichotomy when it comes to groups is the idea of finite groups and infinite groups. If the set is finite, we call it a finite group. If the set is infinite, we call it an infinite group. And with infinite, I mean, we can talk about countable, uncountable, lots of different stuff going on there. But there's this important dichotomy because there are some things true about finite groups that are very different about infinite groups and vice versa. We have seen examples of both types so far. For example, the group Zn with respect to addition is example of a finite group because Zn will have order n. The symmetric group Sn is also an example of a finite group because Sn, the set of permutations, will have order n factorial. The group is itself not order n. Some other examples we saw, there was the dihedral group. This is the set of symmetries of the regular n-gon. We will prove in the future that this group always has order two to the n. There was also the quaternion group, which was a set of eight matrices we introduced in a previous video. This, of course, is a group of order eight. These are all examples of finite groups. We also had the group Zn star. We will show in the future that this, well, this is a finite group, but its order is going to be the totient of n where this is Euler's totion function, which is just the number of relatively prime numbers to n. I mean, this doesn't really need much of a proof there, but we'll talk some more about this in the future right here. These are some examples of finite groups we've introduced already in this series. Some examples of infinite groups, well, I shouldn't say the natural numbers because under addition that's not a group because there's no inverses, but Zqrc under addition, these are all examples of infinite groups. If we look at q star and r star and c star with respect to multiplication, these are still infinite groups even though we threw zero out of the set there. So that's the notion of order. We'll talk a lot about order all the time. Some other conventions we should make mention when we talk about groups. When we're working with a generic group, like it's not a specific group like the ones we just listed. It's just sort of speaking generically. We kind of assume that the group is a multiplicative group. That is, we write the operation as multiplication. So instead of using some general symbol like G circle h or G sharp h are you like G square h or you can see lots of different symbols people use for operations, binary operations. Don't worry about any of those. When it comes to groups, we typically denote them as multiplication in which case with typical multiplication we actually don't use a symbol whatsoever just juxtapositioning the elements next to each other. G h is actually interpreted as the product and that's then the binary operation in our group. Well, if we write our group multiplicatively then that means that the inverse we typically write as G superscript negative one because that's how we can write the inverse multiplicatively. This also leads to exponential notation that if you see something like G to the N where N is just some natural number this just means G times G times G times G times N times. If you take, so like that would make sense if you have a positive integer if you have a negative integer this just means you take the inverse G inverse G inverse G inverse N times. In groups it's common to use the symbol E to represent the identity of a generic group. This is borrowed from the German where the word there starts with an E but also since we're writing multiplicatively it makes sense to just use the number one as the group identity. So you'll often see these conventions that the one or E will represent the identity of the group. I should also mention that if you take an element G to the zero power this is then understood to be the group identity. So these are some, this is some bit of notation we take when we study the general group theories not necessarily specific groups but for general groups we use multiplicative notation to describe them and you'll see that in the forthcoming proofs.