so I noticed that on YouTube there are a lot of videos on the ceelo theorems and in particular how to get an intuitive feeling for how the proofs of the celo theorems work and why they work and what they're important for so I wanted to make a sequence of videos that goes over the proofs of the theorems together with a video or two on examples of using them so as a disclaimer this material requires quite a bit of background so you have to have familiarity with groups Orbitz stabilizers subgroups pretty much a lot of the material that happens particularly in the group Theory section so this video is gonna be dedicated to part one of the ceelo theorem and let's look at the setup to see what we're trying to prove so you're given a group gene that has size P to the K times M where P is a prime and P and M have no common factors so another way to wear this is that P to the K is the highest power of the prime P that divides the size and gene and the theorem is that G is forced to have a subgroup of this size P to decay so let's I actually see why something like this would matter so say you had a group G and a size was 3 squared times 5 squared times 7 squared ok so what this theorem says then is you have some subgroup of size 3 squared which I'll call h1 and then another of size 5 squared which is h2 and another in size 7 squared which is h3 so the idea behind the seal of theorem is that if you want to figure out the structure of G you now have at your disposal three subgroups of really large size that you know live inside of your group G so in this case because they're prime squared size there's a theorem that tells us that these three groups happen to be a billion now there may be many choices for these individual groups but at least we know we can find them and then the goal for finding out how G works is to figure out how these individual groups piece together and then examples of actually using the seal theorems and practice that's the kind of thing that you do you find these sub groups and then figure out how they piece together to figure out what the structure of G is self is so let's go ahead and actually see how this proof starts out and where the proof for this thing comes from so the proof is going to rely on looking at a particular group action on us particular set the set which we're going to call Omega consists of subsets of G of a particular size there are ones of size P to the K all right so this P to the K right over here but we have in our information given so this is not subgroups of size P to the K we don't know yet whether there even is one is this subsets period what we're going to do is construct an action on this particular subset set of subsets and that action is gonna give to light a subgroup of the size that we want so the action is going to be the following so G acts on Omega in the following way so that means that this actually has to take subsets of size P to the K and produce subsets of size P to the K now one thing that you do in group theory that takes elements inside of a group like I said of them and produces a set of the same size is to let the elements act on the original set by multiplication so our action is going to do that is going to take this particular set X and act on it by multiplication so we're going to produce a new set which is GX taken over all the elements X in our particular big set X okay so because the map that sends elements in X to GX is by objective its inverse is multiplication on the left by G inverse this thing at the end will end up having the same size as the original set X so since X has size P to the K this thing on the right actually does have size P to the K so it is in our set here okay now we need to check all of the axioms to this is actually a group action but luckily because we've constructed this by left multiplication by a group element those things are actually quite manageable to realize so I'm not going to prove them but I'm gonna just leave that to you but this is a group action okay so we have a set and a group action we're gonna analyze this group action to hopefully find our subgroup of size P indicate okay so now we have our set and our action doesn't give in for every single G in our group let's analyze this for a bit so first of all let's look at the size of the actual set that we're acting on is so this consists of all the subsets of a certain size the number of elements in our entire group is P to decay so the number the K times M so the number of elements here is the number of elements in the group choose the number of elements were choosing which is P decay okay so one can prove combinatorially that this happens to be congruent to M modulo P the point of this though is that M is co-prime with P so what this means then is if we look at this action of G on Omega the action is going to split into orbits and because the number of elements in omega is this thing right over here one of the orbits is not going to have P as a factor in its size because this is the entire size of the set Omega Omega splits into a disjoint union of orbits the some of those orbit sizes has to be the size of this entire set but that's not a factor of P so we can say then that one of the orbits of this action of G on Omega let's call it O has size not a multiple of P okay so let's pick a particular set in the orbit so the orbit of X that's the entire group acting on X the output has to be O itself because X is in the orbit the way orbits work you have an orbit if you pick something in orbit the orbit of that is that entire orbit okay interesting but this orbit size doesn't have a factor of P in it so if we look at the orbit stabilizer theorem this actually tells us something about the stabilizer of this X so the orbit stabilizers to the theorem says that the size of the stabilizer times the size of the orbit of X has to be the size of the group G itself which is P to the K times M okay but we're saying that this thing here this is the size of this orbit and the size of the orbit is not a multiple of P so we can't have any of this factor right over here in here that means that this piece has to have all of the pinna decay inside of it so we can say that P to the K divides the size of the stabilizer of this particular element we picked in the orbit okay interesting now the stabilizer itself is a subgroup of G it's a set of elements in the group G that under our action fix the set X itself so here now we have a subgroup the stabilizer of X whose size divides this P the decay what we're going to prove is in fact that the size of this stabilizer happens to be exactly peanut decay and consequently this stabilizer which is a sub subgroup of the group G is a group of the size that we're interested in okay so I'm gonna save you some of this data here and go ahead and explore that all right so we found an element of this set so it has size P to decay so that is stabilizer under the action has P to decay as a factor what we're gonna prove is actually that the size of the stabilizer has to be less than or equal to P the decay for a completely other reason and putting these two together the conclusion becomes apparent because we have that the stabilizer has size less than or equal to P to decay but P to the K devices size so these two together would tell you that this stabilizer size is exactly P to decay and since the stabilizer itself is a subgroup of the group gene this implies the stabilizer is a sub group of size P to decay and so it's an example of the type of group that we're looking for so where does this inequality come from this can come from analyzing what stabilizers do themselves so to prove that the size of the stabilizer is less than or equal to P in the can which is the thing that we have left what we're going to do is consider a random element in our particular set X so just to remind ourselves of the framework you have peeed indicate possible choices so there's lots of choices we're gonna pick one of them and we're going to do is consider the map that takes G to itself where we take an element in the group and we map it to G times this a right over here now this thing here is a bijection and the reason is because if we multiply by a inverse here we'll come back to the group itself so definitely a by deduction but what happens to elements in the stabilizer when we perform this so if this thing is in the stabilizer of capital X then that means when we hit it with an element of capital X it stays in capital X so if G is in the stabilizer then this thing is in capital X the outputs of things from the stabilizer of capital X are things that are inside of capital X but this thing is supposed to be a bijection so that means that there's a by ejection between the stabilizer of capital X and some subset of capital X itself so this by ejection on G allows us to show that the size of the stabilizer of X is actually less than or equal to the size of X itself and the size of X itself is P to decay so this thing has size less than or equal to P to decay great so we put everything together we do get indeed that the stabilizer is a subgroup of the size that we want