I'm wondering if homomorphisms $\Phi:GL(2,\mathbb{C})\to \mathcal{M}$ and $\Theta:SL(2,\mathbb{C})\to \mathcal{M}$ are isomorphisms. One can prove that $\Theta$ is sujective, but then $\Phi$ must also be surjetive (since $SL(2,\mathbb{C}) The reason I'm puzzled is because a question I need to solve says that it is $PSL(2,\mathbb{C})$ that is isomorphic to $\mathcal{M}$. So is it true that all of the three groups are in fact isomorphic to $\mathcal{M}$?
$GL(2, \mathbb{C})$ and $SL(2, \mathbb{C})$ isomorphic to Möbius group?
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abstract-algebra
complex-analysis
group-theory
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2Note that $\Theta$ is the restriction of $\Phi$ to a smaller domain. Purely set-theoretically, it is not possible for a proper restriction of an injection to be a surjection (with the same codomain), which automatically means $\Phi$ and $\Theta$ can't *both* be isomorphisms. – 2017-02-02
1 Answers
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No: the homomorphisms are not isomorphisms. In particular, note that any multiple of the identity map (such as $-I$) gets mapped to the identity of $\mathcal M$, which means that both of your homomorphisms fail to be injective, since they have non-zero kernels.
As always, the first isomorphism theorem applies. So since these maps are surjective, we have $$ \mathcal M \cong GL_2/\ker \Phi \cong SL_2 / \ker \Theta $$