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TIFR-2015-Maths-A-10

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For a group $G$, let Aut(G) denote the group of automorphisms of $G$. Which of the following statements is true?

  1. Aut$(\mathbb{Z})$ is isomorphic to $\mathbb{Z}_{2}$
  2. If $G$ is cyclic, then Aut $(G)$ is cyclic.
  3. If Aut (G) is trivial, then $G$ is trivial.
  4. Aut $(\mathbb{Z})$ is isomorphic to $\mathbb{Z}$ 
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