A group with a trivial automorphism group

Let $G$ be a group and suppose $Aut (G)$ is trivial.

Show that $G$ is abelian.
Show that for any abelian group $H$ , the inversion map $ϕ (h) = h^{- 1}$ is an automorphism.
Use parts (1) and (2) above to show that $g^{2}$ is the identity element for every $g \in G$ .

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Let $G$ be a group and suppose $\operatorname{Aut}(G)$ is trivial.
\begin{enumerate}[label=(\alph*)]
	\item Show that $G$ is abelian.
	\item Show that for any abelian group $H$, the {\bfseries inversion map} $\phi(h)=h^{-1}$ is an automorphism.
	\item Use parts (a) and (b) above to show that $g^2$ is the identity element for every $g\in G$.
\end{enumerate}