Finite abelian subgroups of $GL(2,\mathbb C)$ without Pseudoreflection

Question

Let $G$ be a finite abelian subgroup of $GL(2,\mathbb C)$ such that $A-I_2$ does not have rank $1$ for all $A\in G.$ Then, is it true that $G$ is cyclic?

Answer 1

Yes. Note that this induces a representation $G \to GL(2, \mathbb{C})$. Since $G$ is abelian, this is reducible, and the corresponding $\mathbb{C}G$-module can be written as $V \oplus W$ where $V$ and $W$ are one-dimensional.

Since you assumed $A - I_2$ does not have rank 1 for all $A \in G$, we must have that the maps $G \to \mathbb{C}$ are faithful, which is only possible if $G$ is cyclic.