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Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful.

I would like to know whether there is a complete understanding of discrete subgroup of SU(n) and SO(n). If SU(n) and SO(n) are too general, in particular, we can focus on:

(1) all discrete subgroups of SU(2),

(2) all discrete subgroups of SU(3),

(3) all discrete subgroups of SO(3),

(4) all discrete subgroups of SO(4).

I have heard the fact of discrete subgroups of SU(2) is addressed by ADE classification. I also learned from a note about discrete subgroups of SO(3) by Ginzparg's note. But I cannot list down all of the subgroups of SU(2) and SO(3), for example:

(5) whether $(\mathbb Z_2)^3$ or $\mathbb Z_n \times \mathbb Z_m \times \mathbb Z_o$ (with some arbitrary $n,m,o$) are subgroups of SU(2), SU(3), SO(3), SO(4)?

I would be happy to know whether the facts about these groups, such as:

SU$(2)/\mathbb Z_2=$ SO(3)

SO$(4)/\mathbb Z_2=$ SO(3) $\times$ SO(3)

SO$(4) =($SU(2) $\times$ SU(2))/$\mathbb Z_2$

whether these facts can help address the subgroups of SU(2), SU(3), SO(3), SO(4) from one to the other, i.e. if I understand all discrete subgroups of SO(3), can I obtain all discrete subgroups of SU(2) from SU$(2)/\mathbb Z_2=$ SO(3)?

PS. Accidentally I found this paper by Hanany and He, where its Appendix I and III offer table for discrete subgroups of SU(2) and SU(3); however, further illumination on discrete subgroup of SU(n) and SO(n) and my (1), (2), (3), (4), (5) would still be helpful.

Any precise and clear Ref about this discrete subgroups of SU(n) and SO(n) would be helpful. Thank you very much.

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See en.wikipedia.org/wiki/Category:Binary_polyhedral_groups for finite subgroups of $\text{SU}(2)$, which double cover the finite subgroups of $\text{SO}(3)$. It also turns out that $\text{SU}(2) \times \text{SU}(2)$ double covers $\text{SO}(4)$ (but they are not isomorphic as you claim), so the finite subgroups of the latter can be understood in terms of the finite subgroups of the former; this classification is somewhere in Conway and Smith's On Quaternions and Octonions. –  Qiaochu Yuan May 21 '13 at 1:22
    
Thanks YACP for editing, thanks Qiaochu for the comment! –  mystery May 21 '13 at 2:25
    
Note that every finite group is a subgroup of the symmetric group on $n$ letters for some $n$, and so occurs as a discrete subgroup of $O(n)\subset SO(n+1)$, so the complete understanding of discrete subgroups of $SO(N)$ should be pretty hard. –  Dalimil Mazáč May 21 '13 at 16:13

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