 This lecture is part of an online mathematics course on group theory and we'll be covering a few examples of groups, mostly of order 120 or 168. So first we'll look at order 120 which is 2 by 60 and there are plenty of groups of order 120 but we're only going to be interested in the ones that involve the icosahedral group of order 60. And there are four obvious ways to write down an interesting group of order 120. We can take the group SL2 of F5, 2 by 2 matrices of determinant 1 over the field with 5 elements. So the number of elements of this is 5 squared minus 1 times 5 squared minus 5, that's the order of gl2 F5 and then we have to divide it by 4 because we're taking numbers of determinant 1 so this ends up as 120. Or we can take the group that the binary icosahedral group, so you recall that we take the map from the unit quaternions to SO3 of the reels and we take the icosahedral group here and take its endless image here and because this is a double cover, this is twice the order of the icosahedral group which is 2 times 60 which is 120. Thirdly we've got the group S5 of all permutations of five objects and this is order 5 factorial which is equal to 120 and another obvious thing we can do is we can just take 2 times the alternating group A5 which is order 2 times 60 which is also 120. So we've got four groups of order 120 and we can ask which of them are isomorphic to each other. Well one way to sort them out is to look at the Seal of 2 subgroup. So if we work out the Seal of 2 subgroup of these we find here it's the quaternion group, here it's the quaternion group, here it's the dihedral group of order 8 and here it's a product of three groups of order 2. So we see that these three groups are indeed distinct although it's really easy to confuse them because they're all quite similar as we will see. On the other hand these two groups are in fact isomorphic so these are really the same groups as each other. So if you've got a group of order 120 you can ask which of these groups it does it correspond to. For example we can take the group of all symmetries of an icosahedron so I guess again the group of all rotations of an icosahedron has order 60 but you can also do things like reflections of the icosahedron so this is another group of order 120 and if you work out its Seal of 2 subgroup it's z over 2z cubed so we can strongly suspect it's isomorphic to this and in fact it is because the symmetries of an icosahedron has the symmetries plus or minus one and then it has the rotations of the icosahedron so it splits as a product like this so it is in fact 2 times a5. There are three ways to build a group out of a group of order 2 and a group of order 60 you can think of the binary icosahedral group as having the group of order 60 kind of sitting on top of the group of order 2 because the group of order 2 is a normal subgroup this one you can think of the group of order 60 as having the group of order 2 sitting on top of it because the group of order 60 is a normal subgroup and this group they sort of sit side by side because they're both normal subgroups and you remember this is very similar to what happened when we looked at groups of order 24 here we had the binary tetrahedral group that had a group of order 12 with the centre then we had the symmetric group s4 again of order 24 and finally we had the group 2 times the tetrahedral group which sort of you can picture is looking like that so the the tetrahedral group and the icosahedral group kind of behave very very similarly they can both be combined in three interesting ways with a group of order 2. There's a one particular application of these groups is to take the binary icosahedral group so let's take g to be the binary icosahedral group and we know that g is a subgroup of the group s3 of unit quaternions so we can look at the quotient s3 modulo g and this this group here is just a three manifold and this group acts fixed point freely on it so this is a three manifold compact one that's rather famous one called the Poincare homology sphere and now there's you can form the quotient of s3 by any discrete group there are quite a lot of them for instance you can take a cyclic group or the binary tetrahedral group so this is always a three manifold and its fundamental group is easily seen to be just the group h so we can obtain lots of three manifolds with finite fundamental groups so the fundamental group is the first homotopy group and it also has a homology group h1 and the homology group is just the abelianization of the fundamental group pi1 now the group this group that the binary icosahedral group has the property that the abelianization is just trivial this is quite easy to see from the fact that the group a5 is simple so the abelianization must be contained in the center of order two and the abelianization can't be of order two because the binary icosahedral group has no homomorphisms to a group of order two so this group has vanishing first homology and Poincare when he started looking at three manifolds he originally conjectured that if a three manifold has the same homology groups as a three sphere in other words it is vanishing first homology group then it is a three sphere and a bit later he found this counter example well so this group has the same homology as a three sphere but it has different homotopy groups so its first homotopy group is order 120 and he then made the notorious Poincare conjecture that any compact three manifold with the same homotopy groups as a three sphere must actually be a three sphere and this was proved a few years ago by Perlman continuing work of Hamilton as a very difficult result so now we'll discuss the groups of um well a group of order 168 so we've seen that the first simple group has order 160 the next simple group has order 168 and there are two obvious ways of writing down groups of order 168 we can take the group SL3 over a field with two elements and this is order two cubed minus one times two cubed minus two times two cubed minus four which is seven times six times four which is 168 and we can also take the group PSL2F7 so it's a seven not a four and this is order seven squared minus one times seven squared minus seven and then we should divide it by six and then we should divide it by two because we're taking the projective group and this again has order 168 and these two groups are actually isomorphic although this isn't terribly easy to see if you try and write down a homomorphism for one of these groups to the other you'll find it's not at all obvious how to do this I mean you know this is matrices over a field with two elements and it's rather hard to get from that to matrices with fields over groups over a field with seven elements this sort of isomorphism is one of the many things that makes the classification of simple groups rather tricky because there are four groups of small order there are all sorts of accidental isomorphisms between groups for example you might start trying to classify simple groups by trying to work out what their characteristic is because they should be you know some sort of matrix groups over fields of some characteristic and examples like this show that it's kind of tricky to figure out the characteristic of a group because this group has two different characteristics it's a natural matrix group over fields of characteristic two or seven anyway we can write down the some things acted on by this group first is the pharno plane so this is the projective plane over the field of two elements so it's actually done by sl3 of f2 and you can draw a picture of this as follows it's got seven lines and seven points and here are the seven points of the plane and I've also drawn in the seven lines except there are only six of them so I'm going to draw in the seventh line like this so this is actually a straight line in the plane although it looks like a circle because there's there's no way to draw it all with straight lines and if you check the if you look at this graph you can see it's got a lot of automorphisms and if you look carefully you can actually see if it's got 168 automorphisms and is in fact isomorphic to these two groups here another way this group turns off in as this is the automorphism groups of the Klein quadric which is x cubed y plus y cubed z plus c cubed x equals naught so this is a um a quadric that means a degree four curve um inside um the two-dimensional complex plane and this quadric also has autumn automorphism group of order 168 which isn't all that easy to see you can easily see a subgroup of order 21 but getting the remaining automorphisms is kind of difficult um this is an example of a Howitz surface so if you've got a compact Riemann surface s of genus g greater than one then Howitz showed that the automorphism group of s has order at most 84 times g minus one um and there are no examples for g equals two and for g equals three there is this example here so for this surface has genus g equals three and its automorphism group is exactly equal to 84 times three minus one which is 168 and it's the unique Riemann surface of genus three with an automorphism group this large um so uh i think that's all i want to say about these groups what we'll do next lecture is discuss simple groups in a bit more detail and give the Jordan Holder theorem saying that groups can be decomposed into simple groups and the number of simple groups that occurs is unique