| [0.00s -> 12.03s] This video will be a summary based on all the videos in this series, so please watch all these videos first. But here is a quick recap on the concepts in all these videos. |
| [12.03s -> 24.78s] In the first chapter, instead of using the standard definition using axioms, we introduce groups as symmetries of objects, which lays the foundation for the entire video series. |
| [29.26s -> 43.60s] as well as making the concept of group actions easier to understand, because symmetries naturally act on the vertices, edges or faces. Then, we derive the orbit stabilizer theorem |
| [46.03s -> 55.98s] which states that the size of the orbit multiplying the size of the stabiliser of any element in the set X is the size of the group G. |
| [55.98s -> 64.58s] Next, we discover the fact that the size of the stabiliser divides the size of the group can be argued in a different way. |
| [64.58s -> 79.50s] And then we extract the properties of the stabiliser we used to define what a subgroup is. Then we discover Lagrange's theorem, which states that the size of a subgroup divides the size of the whole group. |
| [80.50s -> 90.38s] Then we introduce the concept of conjugation, and figure out its intuition to be just viewing the symmetry being conjugated in another perspective. |
| [90.38s -> 96.14s] This allows us to introduce the concept of normal subgroups and simple groups. |
| [97.65s -> 107.82s] which in turn allows us to define what a quotient group is, which is a group of the cosets where we can define a consistent operation. |
| [114.70s -> 120.35s] The video also consisted of seeing the concept in different perspectives. |
| [120.35s -> 130.48s] In the sixth chapter, we then introduce the concept of homomorphism, and then understand the three statements of the isomorphism theorem more intuitively. |
| [137.46s -> 151.60s] Then finally in the last chapter, we discover that group actions can be thought of as a homomorphism. And we also prove Cayley's theorem as a direct application of this change of perspective. |
| [156.53s -> 165.23s] With all these recaps done, now we are going to see an example that will use all the concepts that we have seen before. |
| [165.33s -> 178.98s] Suppose G is a simple group with 60 elements, and H is a subgroup of G. By considering an action of G on the coset of H, show that H cannot have two |
| [178.98s -> 183.86s] three of four cosets. Let's dissect what this means. |
| [183.86s -> 197.71s] G being a simple group means that its only normal subgroups are the identity and the group G itself. So whatever we need to do, it would somehow be related to normal subgroups. |
| [197.71s -> 210.35s] groups. Next, we have the number 60. But the only concepts that we discussed that will involve some sort of numbers are the orbit stabilizer theorem and the Lagrange's theorem. |
| [210.35s -> 221.81s] because they refer to products of something. For the next line, we just note that subgroup is a concept that we have seen in the third chapter. Nothing too special here. |
| [221.81s -> 230.24s] The next line is quite special, because it specifically tells us what to do. So this should be a useful starting point. |
| [230.24s -> 242.58s] Just as a reminder of what a coset is, we have this mental picture of the subgroup and its coset, so the cosets are just the rectangles tiling up the entire group G. |
| [242.61s -> 256.75s] The last line is what we need to prove. And this result is not immediately obvious either. Apparently, it must have something to do with simplicity, and the number 60 somehow. And we are going to figure out |
| [256.75s -> 270.77s] how these things are connected. The third line provides us with a starting point, and the line of reasoning would be, what if G is simple with 60 elements, but H has 2, 3 or 4 cosets? |
| [270.77s -> 283.44s] it should have some contradiction with the theorems that we know, then we can show that H cannot have 2, 3 or 4 cosets. Pause the video if you want to have a go yourself. |
| [283.44s -> 293.71s] The first step is to consider an action of G on cosets of H, which sounds abstract, but the action we will be considering is very straightforward. |
| [293.71s -> 303.76s] Again, we consider this mental picture we have for this more abstract action. Each dot represents a coset, including the subgroup H. |
| [303.76s -> 312.62s] Considering the action of G on these cosets, this simple action will transform any coset to some other coset. |
| [312.62s -> 319.41s] This action will send H to the coset GH, G inverse H to H itself, and so on. |
| [320.69s -> 334.42s] So we now have a group action of G on the cosets of H, given by this relation where you just stick a G1 in front. We will then have the corresponding homomorphism, say phi. |
| [334.42s -> 345.02s] from the group G to the symmetric group on the cosets of H. This opens up the possibility of analysing the group action with the isomorphism theorem. |
| [345.02s -> 353.98s] The isomorphism theorem states three things once we have a homomorphism between two groups. Let's look at the first statement here. |
| [353.98s -> 367.04s] It is given that G is simple, which means that the kernel can only be the identity set or the group G itself, because these are the only normal subgroups that G has. |
| [367.04s -> 372.06s] Now consider the case where this kernel is the entire group G. |
| [372.06s -> 386.69s] Because the kernel is everything that gets mapped to the identity, and in this case, the identity of the symmetric group of the cosets of H would be the identity permutation, which is the permutation where everything stays fixed. |
| [386.69s -> 396.75s] If the kernel is the entire group G, then for all G in the group, the corresponding permutation would look like this, not moving anything. |
| [396.75s -> 409.90s] To show this is not possible, consider G acting on H. H would be sent to GH. Now this isn't done yet, because sometimes GH would be the subgroup H itself, |
| [409.90s -> 415.41s] like when G is actually a member of the subgroup H. But this cannot happen for |
| [415.41s -> 427.60s] all G inside the entire group, because the only case when all elements of the group G will send H to itself is that there is only one coset to begin with. |
| [427.60s -> 440.99s] Since our line of reasoning is to assume that H has two, three or four cosets and see what happens, this case when kernel of V is the entire group G is not possible. |
| [440.99s -> 452.77s] Back to where we were a minute ago. All the discussion just now shows that the kernel can only be the identity set. Next up, we can use the second statement here. |
| [452.77s -> 464.29s] Since it uses the concept subgroup, we are going to use Lagrange's theorem, which states in this context that the size of the image divides the size of the symmetric group. |
| [464.29s -> 476.62s] Lastly, from the last statement, all we need to use is the fact that isomorphic groups must have the same size, because the isomorphism has to be 1 to 1. |
| [476.78s -> 488.56s] then recall from the quotient group video where I explained why it is called a quotient group. The size of the quotient group is just a quotient of the sizes. |
| [488.56s -> 501.74s] this also just comes directly from Lagrange's theorem. So that means in the last statement, we have the size of the quotient group to be the size of G divided by the size of kernel. |
| [501.81s -> 515.81s] Since the group has size 60, and the kernel is just the identity set, the size of the image can then be just calculated as 60. So in this second implication that we draw, |
| [515.81s -> 525.39s] we can replace the size of the image as 60. So far everything is good, but this implication here will be the main punchline. |
| [525.39s -> 538.90s] Let's rewrite this bit here at the top. The only mystery that we need to tackle is the size of the symmetric group. Let's do this in general. If we want to find out the size of the symmetric group of n things, |
| [538.90s -> 552.08s] then the first thing under consideration can be put in n different places, including the option to not move. Then, the second thing can be put in n-1 different places, and so on. |
| [552.08s -> 562.18s] This means that the size of the symmetric group, that is, all the permutations of n things, would be n times all the way to 1. |
| [562.18s -> 574.08s] because the first thing has n places for choice during permutation, and the second thing would have one fewer choice, and so on. So we apply the result we just discovered. |
| [574.08s -> 585.97s] If we have only two cosets of H, the size of the symmetric group would be 2 times 1 equals 2. This is not possible because 60 does not divide 2. |
| [585.97s -> 599.74s] What if there are 3 code sets of H? The symmetric group would then have size 6. Not possible. In a similar fashion, if there are 4 code sets, 60 still does not divide the size of the symmetric group. |
| [599.74s -> 603.95s] which completely proves the result that we need to show at hand. |
| [603.95s -> 616.70s] I know that this proof is rather involved because it covers most of the concepts that we have covered in this video series, so feel free to replay the entire proof and pause when necessary. |
| [616.70s -> 627.89s] If you are ready for some more, it can be shown that there is only one group up to isomorphism that is simple and has 60 elements. The proof of this is way |
| [627.89s -> 639.18s] way beyond the scope of this video series, but we can show what this group structure looks like. It is precisely the group of rotational symmetries of an icosahedron. |
| [639.18s -> 653.26s] To link to really all the concepts we have covered, try using the orbit stabiliser theorem to see why this group has 60 elements, and use the intuition of conjugation to see why this group is simple. |
| [653.26s -> 666.27s] Great thanks to all your support to this group theory video series, and I really hope that you can grasp the intuition to aid your learning, whether you are studying or about to study this interesting field of mathematics. |
| [666.27s -> 673.33s] If you enjoyed this video, give it a like and subscribe to the channel with notifications on. See you next time! |
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