A remark On Abelianized Absolute Galois Group of Imaginary Quadratic Fields

The main purpose of this paper is to extend results on isomorphism types of the abelianized absolute Galois group ${\mathcal{G}}_K^{ab}$, where $K$ denotes imaginary quadratic field. In particular, we will show that if the class number $h_K$ of an imaginary quadratic field $K$ different from $\mathbb{Q}(i)$, $\mathbb{Q}(\sqrt{-2})$ is a fixed prime number $p$ then there are only two isomorphism types of ${\mathcal{G}}_K^{ab}$ which could occur. For instance, this result implies that imaginary quadratic fields of the discriminant $D_K$ belonging to the set $\{-35,-51,-91,-115,-123,-187,-235,$ $ -267,-403, -427 }$ all have isomorphic abelian parts of their absolute Galois groups.