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Is there any reference which classifies the finite subgroups of $SU_2(C)$ up to conjugacy ?

What I know is that lifting the finite subgroups of $SO_3(R)$ by the map $SU_2(C) \rightarrow PSU_2(C)$ gives rise to the groups : cyclic groups of even order, dicyclic groups, binary tetrahedral/octahedral/icosahedral groups. But I dont know if they are the only one.

Thomas Connor
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user10676
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3 Answers3

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The image of a finite subgroup of $\text{SU}(2)$ in $\text{SO}(3)$ is a finite subgroup of $\text{SO}(3)$; moreover, the kernel is either trivial or $\{ \pm 1 \}$. But $-1$ is the unique element of order $2$ in $\text{SU}(2)$, so any group of even order contains it.

I claim all the finite subgroups of odd order are cyclic. This follows because the inclusion $G \to \text{SU}(2)$ cannot define an irreducible representation of $G$ (since otherwise $2 | |G|$), hence it must break up into a direct sum of dual $1$-dimensional representations.

So once you know the finite subgroups of $\text{SO}(3)$, you already know the finite subgroups of $\text{SU}(2)$.

Qiaochu Yuan
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Here is a full list of (closed) subgroups of $ SU_2 $.

There are three maximal subgroups (up to conjugacy). The first maximal subgroup, $ N $, has dimension $ 1 $ and has two connected components $$ N= \{ \begin{pmatrix}e^{i\theta}&0 \\ 0 & e^{-i\theta} \end{pmatrix} 0\leq\theta<2\pi \} \cup \{\begin{pmatrix} 0 & ie^{-i\theta}\\ ie^{i\theta}&0\end{pmatrix} 0\leq\theta<2\pi \} $$ $$ =<U_1,i \begin{pmatrix} 0 & 1\\ 1&0\end{pmatrix}> $$ and contains subgroups $$ U_1=\left \{\begin{pmatrix}e^{i\theta}&0 \\ 0 & e^{-i\theta} \end{pmatrix} \mid 0\leq\theta<2\pi \right \} $$ and all cyclic groups of order $ n $ $$ C_n=\left \{\begin{pmatrix}e^{2 \pi k i/n}&0 \\ 0 & e^{-2 \pi k i/n} \end{pmatrix} \mid 0 \leq k < n \right \} $$ and all binary dihedral groups of order $ 4n $ \begin{align*} {BD}_{n}=&\left \{\begin{pmatrix}e^{2 \pi k i/2n}&0 \\ 0 & e^{-2 \pi k i/2n} \end{pmatrix} \mid 0 \leq k < 2n \right \} \\ & \cup \left \{\begin{pmatrix} 0 & ie^{-2 \pi k i/2n}\\ ie^{2 \pi k i/2n}&0\end{pmatrix} \mid 0 \leq k < 2n \right \}\\ =&<C_{2n},i \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}> \end{align*} Note that we have SES $$ 1 \to U_1 \to N \to C_2 \to 1 $$ which look like the SES for $$ 1 \to U_1 \to O_2(\mathbb{R}) \to C_2 \to 1 $$ But these groups are distinct. In particular the only element of $ SU_2 $ with order $ 2 $ is the matrix with eigenvalues $ -1,-1 $ $$ \begin{pmatrix} -1 & 0\\ 0&-1\end{pmatrix} $$ by contrast $ O_2(\mathbb{R}) $ has infinitely many elements of order $ 2 $. It is also interesting to note that $ U_1 $ is the maximal torus of $ SU_2 $ and that $ N $ is $ N(T) $ the normalizer of the maximal torus. So the short exact sequence above is really $$ 1 \to T \to N(T) \to W \to 1 $$ where $ T $ is the maximal torus, $ N(T) $ is the normalizer and $ W $ is the Weyl group (the Weyl group for $ SU_2 $ is symmetric group on two letters, equivalently two element cyclic).

The two other maximal subgroups are finite: the binary icosahedral group $ 2.A_5 \cong SL(2,5) $ of order $ 120 $ and the binary octahedral group $ 2.S_4^- \cong SL(2,3):2 $ of order $ 48 $, which has GAP ID $SmallGroup(48,28)$. It is the schur cover of $ S_4 $ of minus type. A word of warning: The group $ SmallGroup(48,29)\cong GL(2,3)\cong 2.S_4^+ $ is a subgroup of $ U(2) $ which has also has order $ 48 $ and a fairly similar looking character table, it is the schur cover of $ S_4 $ of plus type. But there is no subgroup of $ SU(2) $ isomorphic to $ GL(2,3) $. Another way in which these groups are closely related is that $ 2.S_4^- $ and $ 2.S_4^+$ are both subgroups of $ U(2) $ and their intersection is the binary tetrahedral group $ 2.A_4 $ and every element of $ GL(2,3) \setminus SL(2,3) $ is just $ i $ times an element from $ 2.S_4^- \setminus SL(2,3) $ (that explains why half the elements have determinant $ -1 $ and so $ GL(2,3) $ is not contained in $ SU(2) $. Comparing $ GL(2,3) $ and the binary octahedral group

The intersection of these two groups is exactly $ 2.A_4 \cong SL(2,3) $ the binary tetrahedral group of order 24. These three binary polyhedral groups are the three exceptional subgroups of $ SU_2 $. They are the only primitive subgroups. They are also all Lie primitive (not contained in any proper positive dimensional closed subgroup).

To reiterate, the closed subgroups of $ SU_2 $ are classified as follows:

The finite subgroups are given by the ADE classification with $ A_n $ corresponding to cyclic order $ n $, $ D_n $ corresponding to binary dihedral group of order $ 4n $, $ E_6 $ corresponding to binary tetrahedral, $ E_7 $ to binary octahedral and $ E_8 $ to binary icosahedral.

There are only three closed subgroups of positive dimension: $ SU_2 $ itself, the maximal torus $ U_1 $, and $ N=N(T) $ the normalizer of the maximal torus.

Ian Gershon Teixeira
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  • Excellent answer, @Ian. Is there a resource (handbook or website) that lists subgroups for various classical groups? Wikipedia isn't quit there yet... – Travis Lee Mar 02 '23 at 08:17
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    https://arxiv.org/abs/math/0605784 is not a bad place to start. Is there something in particular you're looking for? – Ian Gershon Teixeira Mar 02 '23 at 14:44
  • Thanks, @Ian. My level is roughly Lie Algebra 101, now self-studying 102. That paper probably contains it all, though it is not very accessible. Table 5 lists non-simple subgroups, but not simple, which are mentioned briefly for the Lie Algebras. If I were to write out the explicit case of SU(3) from the general expressions, I don't trust myself to get it 100% right. I'm surprised that I cannot find handbook or online wiki that lists these things explicitly, as you did for SU(2) in your answer. – Travis Lee Mar 03 '23 at 10:09
  • @TravisLee If you are interested in (closed) subgroups of $ SU(3) $ then you might like https://mathoverflow.net/questions/65522/lie-subgroups-of-su3 or https://math.stackexchange.com/questions/497853/closed-lie-subgroups-of-su3 the highlights are that the two big connected subgroups are $ U(2) $ and $ SO(3) $. You already know about the subgroups of those two groups. Also $ T^2 \rtimes S_3 $ is the normalizer of the maximal torus. Then there are some interesting finite subgroups which I discuss in my answer above. If you post a question about classifying subgroups of $SU(3)$ I'll look at it. – Ian Gershon Teixeira Mar 03 '23 at 14:16
  • your links have been helpful. I notice others asking, like me, because they don't readily find resources listing group properties, such as subgroups! – Travis Lee Mar 06 '23 at 17:18
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Yes, the ADE classification enumerates all possible finite subgroups of $SU(2)$. Just a trivial correction, when it comes to the Abelian cyclic groups, A, the order may be both even and odd because $U(1)$ inside $SU(2)$ has all these $Z_N$ subgroups. The odd ones aren't linked to subgroups of $SO(3)$ in the same way.

The dicyclic groups, D, and the three exceptions, E, are as you wrote.

Luboš Motl
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