Let G be an infinite cyclic group. Prove that Aut(G)≅(Z₂,⊕).

2. Hint1: $G\simeq \mathbb{Z}$
Hint2: If $G_1\simeq G_2$ then $\mbox{Aut}(G_1)\simeq \mbox{Aut}(G_2)$.

3. Originally Posted by hzhang610
Let G be an infinite cyclic group. Prove that Aut(G)≅(Z₂,⊕).
Any homomorphism from $\mathbb{Z}$ to itself is of the form $n\to kn$ for some $k\in\mathbb{Z}$. This is a bijection only if $k=\pm1$. Hence there are only two automorphisms of $\mathbb{Z}$.