what is the action of $\operatorname{Aut}(\mathbb{Z})$ on $\mathbb{C}^n$

Question

Calculate the following actions:

$(i)$ Find an action of the group $GL(n,\mathbb{R})$ on the Euclidean space $\mathbb{R}^n$.

$(ii)$ Find an action of the group $GL(n,\mathbb{C})$ on the Euclidean space $\mathbb{C}^n$.

$(iii)$ Find an action of the group $\operatorname{Aut}(\mathbb{Z})$ on the Euclidean space $\mathbb{C}^n$.

Answer:

$(i)$

Let $A \in GL(n, \mathbb{R})$ and $ v \in \mathbb{R}^n$, then the action is defined by $A \cdot v=Av$

i.e, the group act as a linear transformation.

$(ii)$

The similar action can be described in this case also.

$(iii)$

We know $ \operatorname{Aut}(\mathbb{Z})=\{1,-1\}$.

Also the automorphism group $Aut(G)$ acts on $G$ by $ \psi \cdot g=\psi(g), \ \ \psi \in Aut(G)$ and $g \in G$.

But I do not know what is the action of $\operatorname{Aut}(\mathbb{Z})$ on $\mathbb{C}^n$ .

Is it Frobenius action or something else?

Kindly explain and answer this part $(iii)$.

Answer 1

Since $\operatorname{Aut}(\mathbb Z)\simeq\{1,-1\}$, you can make it act on $\mathbb C^n$ by making $1$ act as $\operatorname{Id}$ and $-1$ act as $-\operatorname{Id}$, for instance.