What is meant by the notation complex numbers with superscript cross: $\mathbb{C}^{\times}$

Question

The context I'm seeing it in says it's isomorphic to the automorphisms on $\mathbb{C}$. Thanks for your help.

Answer 1

Given a field $\mathbb F$, the notation $\mathbb F^{\times}$ denotes the multiplicative group of the field. That is, it is the group where the space is the non-zero elements of $\mathbb F$, and the binary operator is what, in the field, is the multiplicative operator (zero has to be excluded because zero does not have a multiplicative inverse). The multiplicative group of a field is always isomorphic to a (not necessarily proper) subgroup of the automorphism group of the field; this can be shown by the fact that $\phi: a \rightarrow (f_a:b \rightarrow ab)$ is a homomorphism (note that $\phi$ is a function whose input is an element of $\mathbb F$, and output is itself a function from $\mathbb F$ to $\mathbb F$).

I'm not clear on how $\mathbb C^{\times}$ is isomorphic to the set of automorphisms; it seems to me that $\{f_a:b \rightarrow ab \}$ and $\{a \rightarrow a, a \rightarrow \bar a \}$ are both subgroups of $\text {Aut}(\mathbb C)$. Perhaps the claim is that $\mathbb C^{\times}$ is isomorphic to the orientation-preserving automorphisms?

Answer 2

This refers to the complex numbers without $0$. The automorphism is given by multiplication.