TITLE: How is $\operatorname{GL}(1,\mathbb{C})$ related to $\operatorname{GL}(2,\mathbb{R})$?
QUESTION [19 upvotes]: I am trying to get a grasp on what a representation is, and a professor gave me a simple example of representing the group $Z_{12}$ as the twelve roots of unity, or corresponding $2\times 2$ matrices. Now I am wondering how $\operatorname{GL}(1,\mathbb{C})$ and $\operatorname{GL}(2,\mathbb{R})$ are related, since the elements of both groups are automorphisms of the complex numbers. $\operatorname{GL}(\mathbb{C})$, the group of automorphisms of C, is (to my understanding) isomorphic to both $\operatorname{GL}(1,\mathbb{C})$ and $\operatorname{GL}(2,\mathbb{R})$ since the complex numbers are a two-dimensional vector space over $\mathbb{R}$. But it doesn't seem like these two groups are isomorphic to each other.

REPLY [14 votes]: The one group is commutative, the other is not. So they can not be isomorphic.