Group Theory

8 Finite subgroups of SU(2), SO(3)

Let’s try to classify the finite subgroups of \(\SU (2)\), which consists of matrices of the form \(\begin {pmatrix} a & b \\ -\overline {b} & \overline {a} \end {pmatrix}\) where \(|a|^2+|b|^2 = 1\). Note that \(\SU (2)/\{\pm 1\} = \SOr (3)\), so every finite subgroup is either a subgroup of \(\SOr (3)\) or the pullback of one. One family of subgroups of \(\SOr (3)\) is \(C_n\), and another is \(D_n\). We can get three more \(E_6,E_7,E_8\) by looking at the symmetries of the platonic solids. \(E_6\) comes from the tetrahedron, \(E_7\) comes from the cube/octahedron, and \(E_8\) comes from the dodecahedron and icosahedron. It is easy to compute their orders by observe that they act simply and transitively on the oriented edges of the tetrahedron, octahedron, and icosahedron, so the orders are \(12,24,60\) respectively. In fact they are familiar groups. In particular, note that \(E_6\) acts faithfully on the faces, of which their are \(4\), so it is easy to see it is \(A_4\). \(E_7\) acts faithfully on pairs of opposite vertices of the cube, so it is \(S_4\). One can group the edges of the dodecahedron in \(6\)s such that \(E_8\) acts faithfully on its \(5\) translates, showing that \(E_8 = A_5\). \(E_6,E_7,E_8\) are also also named after their platonic solids (eg: \(E_8\) is the icosahedral group).

To classify all subgroups, for a finite subgroup \(G\), consider its action on the unit sphere. The quotient is a surface, and in fact must be a sphere by topological considerations (eg. cohomology). Then by putting a holomorphic structure on the quotient, the quotient map \(\pi \) turns into a branched cover. By looking at Euler characteristic, \(\frac 2{|G|} = 2-\sum _{p}(1-\frac {|\pi ^{-1}(p)|}{|G|})\). Thus there can be at most \(3\) ramification points, and so \(G\) has three generators \(a,b,c\), with relations \(abc = 1,a^j,b^k,c^l\) corresponding to the possible ramification points (\(j,k,l\) are chosen minimally and may be \(1\)). Now \(j,k,l\) must be \(\frac {|G|}{|\pi ^{-1}(p)|}\) for their respective \(p\) since the action is Galois (look at monodromy). Now we can take the Galois cover corresponding to \(\langle a,b,c|a^j,b^k,c^l,abc\rangle = D(j,k,l)\), the von Dyck group, in the complement of the ramified points. By our Euler characteristic argument, \(1+\frac 2{|G|} = \frac 1 j+\frac 1 k + \frac 1 l)\). Note that for the \(j,k,l\) that could possibly make this hold, this group is finite. Then, note that this covers our degree \(|G|\) cover, and is unramified, hence we can fill it in to give an unramified covering of \(S^2\), which is an isomorphism as it is simply connected. Thus \(G = D(j,k,l)\). \((j,k,l) = (2,2,n),(1,n,n),(2,3,3),(2,3,4),(2,3,5)\) work and give us the groups we already found. In order for the equality to hold, it is simple to check that these are the only possibilities. When looking back at \(\SU (2)\), we get central extensions, of these groups by \(\pm 1\), and we add the word binary to indicate this. For example, the preimage of the icosahedral group is the binary icosahedral group. The only central extensions that split are the cyclic groups, so we have classified subgroups of \(\SU (2)\) and \(\SOr (3)\).