$GL(2, \mathbb{C})$ and $SL(2, \mathbb{C})$ isomorphic to Möbius group? 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 $SL(2,\mathbb{C})<GL(2,\mathbb{C})$). But it seems obvious (and easy to prove) that both $\Phi$ and $\Theta$ are injective as well. So $\Phi$ and $\Theta$ must be isomorphisms.
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 $\mathcal{M}$?

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
$$
\mathcal M \cong GL_2/\ker \Phi \cong SL_2 / \ker \Theta
$$
