Group Theory -- Simple automorphism

[snow ice]

I am trying to show that the order of $\mathrm{Aut}(\mathbb Z, +)$ is 2, and I've got that if $\varphi \in \mathrm{Aut}(\mathbb Z)$, and $g \in \mathbb Z$, then $\varphi (\langle g \rangle) = \varphi (g \mathbb Z) = g \mathbb Z$, but that last step feels out of place... What step do you guys think I missed ?

[radiant solstice]

ταυταυ

[pastel dagger]

Aut(Z,+) is the set of all automorphisms on Z. You are going to need to consider properties of automorphisms on a cyclic group (Z,+ is cyclic) to find out all possible automorphisms.

[pastel dagger]

Or in other words, check the generators and see what possible automorphisms can do to generators.

[snow ice]

Yeah

Sorry I did not come back

But I solved it

Since like you said I looked at the generator and

<1> = <-1>

Meaning we can map x |-> x and x |-> -x in oversimplifications

.solved