 This last video is gonna be short. I'm gonna present you one more proposition. This one's not gonna go with proof this time. The proof is somewhat elementary and left as an exercise to the viewer here. In a group, the usual exponential rules hold. That is, if you have general elements, G and H inside your group and you have integers M and N, the following two things hold. If you take GM times GN, this will equal G to the M plus N. You can add the exponents together here. And the basic idea is if you have M many G's and then you have N many G's, we can put these together. You have M over here, you have N over here. This will combine to give us N plus N. But of course in order for G to the M to even make sense, you need associativity because just take G cubed for example, right? Is G cubed GG times G or is it G times GG? Which one is it? Well, by associativity, it doesn't matter. And therefore we get that G squared, G is the same thing as G squared G. And so this actually is kind of an interesting observation right here. What you see is that clearly the screen right here is that if you take powers of an individual element, they necessarily have to commute because G to the M, G to the N is equal to the GM plus N, which is the same thing as G to the N plus M, which is the same thing as G to the N times G to the M. So although there's not necessarily universal commutivity in a group, in some respect there has to be some level of commutivity. There's gonna be some elements that commute because of the associativity law. Now the proof of property A right here actually is a argument using double induction on the associativity there. You have to induct on M while N is fixed and then you induct on N. And so I'm not gonna worry about that double induction argument. I'll let you think about that a little bit more. Another one that follows by the same sort of reasoning use the associativity axiom plus double induction. If you take G to the M and you do that N times, that's gonna be G to the MN. The element, the naive proof, right? This isn't a formal proof, right? But the naive proof is the following. You have G to the M times G to the M times G to the M and you do this of course N times, right? You have M here, M here, M here, M here. You're gonna get N many M's and therefore you get N times M. Again, we're not gonna go through all of these. But there's a double induction argument that works there. But again, this is just the same thing as G to the MN, which is the same thing as G to the N times G to the M. So these usual exponential rules hold here. Now you have to be a little bit careful when you take negative exponents because negative exponents represent inverses here. And also if you take, you also have to be careful if you take a product to an exponent, right? G H to the N. This is not necessarily the same thing as G to the N H to the N. This will be true if you have an Abelian group, but how does one get from here to here in general? The usual argument looks something like the following. You have G H, G H, G H, and you do this, you know, N times. The idea is all gather all the G's together and then gather all the H's together but that requires commutivity. It requires we all to, well it uses associativity, but it also requires commutivity which you don't have in general. So if we have an Abelian group, we can distribute exponents across division, across multiplication. But how does it work in general? G H to the N, we can write that as H inverse, G inverse to the negative N, but that's about all we can do. This is just the Shusach principle written in an iterative process, right? Now speaking about Abelian groups, if you do have an Abelian group, it's very common to write the notation actually additively as opposed to multiplicatively. You can do it multiplicatively, that's fine. But in group theory, whenever you write a group in additive notation, additive notation will always mean an Abelian group. You never, ever, ever use addition to represent a non-commutative group. It's like a carbonyl syn. I actually read this in Joseph Rothman's algebra textbook and he actually got exiled to the island of Patmos for doing that. There was a section where he was writing a non-Abelian group using the operation of addition and he'll never be the same for such a thing. And so Abelian groups, we sometimes use additive notation. We can use multiplicative notation, it's fine. But when you see additive notation, that will always mean Abelian group. It always means commutivity in that context. Now, if you're writing the group additively, the operation will look like G plus H. The inverse, you don't write it as G inverse, you actually would write it as negative G, right? Because subtraction's the inverse of multiplication. And when you iterate the process, when you iterate your product, it really should be a sum. So we think of it not as G to the end, but we think of it as N times G because it's G plus G plus G plus G, right? And the inverse would be negative NG. So negative G plus, negative G plus, negative G plus, negative G. You just make all the appropriate modifications to it. In terms of the group identity, you typically call it zero instead of one, but you could use an E if you want to. That's commonly done as well. The exponential rules we saw right above, they're now off the screen. If we rebranded those for Abelian groups, you'd see the following. If you iterate G and you iterate M times, and you iterate G N times, and you combine that together, that should be iterating G M plus N times. But when you write this in additive notation, this exponential rule actually looks like the distributive property here. And after all, the exponential rules you saw before is because exponents is an iteration of multiplication. Well, with real numbers, multiplication is supposed to be an iteration of addition. And so the distributive property that we know for real numbers is really just a consequence that multiplication iterates addition. And so the exponential laws in additive group look like the distributive property. And when you look at this one, if you do G N times and then you do all of those M times, that's the same thing as M N G. This exponential law, when you write in additive notation, realizing that multiplication is just iterated addition, this actually looks like an associativity law. And so associativity of usual multiplication, real number of multiplication, really just consequence of associativity of addition, believe it or not. And then the principle we saw before, right, the last one that only worked really for Abelian groups. If you have two different elements, G plus H, and you do it M times, you can actually do G M times H M times and combine them together. So this is the other distributive law that also holds for additive groups. So it's kind of interesting when you think about it this way that addition with real numbers forms a group, which means the operation is associative. And therefore, we can prove things by like the distributive laws, associativity of multiplication because addition itself is associative and multiplication just iterates addition. So in some regard, you can build multiplication for real numbers from addition of real numbers. But this is really getting us far afield pun intended there because we're talking about rings and fields right now, which is a topic we will talk later about in this series. But that'll then conclude our lecture nine about properties of groups. These are properties that are true for any group whatsoever. And with the exceptions of things like this, we see that most of these properties, there's just the one property, honestly, all of this was proven without the commutivity of the group. We don't actually need an Abelian group for many of these properties. Associativity, inverses and identities is sufficient.