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