I ask a question about finite subgroup of $GL(2,\mathbb{C})$.
I want to classfy the finite subgroup of $GL(2,\mathbb{C})$
(especially, subgroups of order $p^3$ , $p$ is prime number.)
Is it possible to classfy?
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1The list of finite subgroups of that group is well-known. Google should find you the list and the proofs easily! Try "finite subgroups of gl(2,c)" – Mariano Suárez-Álvarez Feb 26 '17 at 17:54
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What is the word that I should search using goolge? – log log Feb 26 '17 at 18:00
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"finite subgroups of gl(2,c)", unsurprisingly (wrap the words between quote marks) – Mariano Suárez-Álvarez Feb 26 '17 at 18:02
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Is this information from google search only SL(2,C)? Of course, it contains SL(2,C). But, I think it doesn't contain GL(2,C). – log log Feb 26 '17 at 18:16
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See this MO-question for references. – Dietrich Burde Feb 26 '17 at 19:03
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It follows from the arguments in the answers to your other question that a subgroup of $GL_2(\Bbb{C})$ of order $p^3$ is either abelian, or $p=2$. Is that clear to you, or do you want it detailed? – Jyrki Lahtonen Feb 27 '17 at 07:23
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I want details just in case. – log log Feb 27 '17 at 11:59
1 Answers
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If $G$ is a finite subgroup of $\rm{GL}(2,\mathbb{C})$ then we can state this as there is a homomorphism (representation) $\rho:G\rightarrow \rm{GL}(2,\mathbb{C})$.
It is a standard result that this representation we can look into subgroup $U(2,\mathbb{C})$ of unitary matrices (See, Artin's Algebra).
So, your question reduces to finding finding finite subgroups of $U(2,\mathbb{C})$. This has been well studied.
For example, in Complex Functions by Jones and Singerman, the beginning chapter is devoted to study of finite subgroups of this group. Since the complete proof is very long, it is appropriate here just to mention its source.
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