What are the finite subgroups of $SU_2(C)$?

Is there any kind of reference which identifies the limited subgroups of $SU_2(C)$ approximately conjugacy?

What I recognize is that raising the limited subgroups of $SO_3(R)$ by the map $SU_2(C) \rightarrow PSU_2(C)$ generates the teams: cyclic teams of also order, dicyclic teams, binary tetrahedral/octahedral/icosahedral teams. Yet I do not recognize if they are the just one.

2022-06-07 14:36:50
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Answers: 2

Yes, the ADE category identifies all feasible limited subgroups of $SU(2)$. Simply an unimportant improvement, when it involves the Abelian cyclic teams, A, the order might be both also and also weird due to the fact that $U(1)$ inside $SU(2)$ has all these $Z_N$ subgroups. The weird ones aren't connected to subgroups of $SO(3)$ similarly.

The dicyclic teams, D, and also the 3 exemptions, E, are as you created.

2022-06-07 15:08:01

The photo of a limited subgroup of $\text{SU}(2)$ in $\text{SO}(3)$ is a limited subgroup of $\text{SO}(3)$ ; in addition, the bit is either unimportant or $\{ \pm 1 \}$. Yet $-1$ is the one-of-a-kind component of order $2$ in $\text{SU}(2)$, so any kind of team of also order has it.

I assert all the limited subgroups of weird order are cyclic. This adheres to due to the fact that the incorporation $G \to \text{SU}(2)$ can not specify an irreducible depiction of $G$ (given that or else $2 | |G|$), therefore it has to separate right into a straight amount of twin $1$ - dimensional depictions.

So as soon as you recognize the limited subgroups of $\text{SO}(3)$, you currently recognize the limited subgroups of $\text{SU}(2)$.

2022-06-07 15:05:29