Elements of finite order

Let $G$ be an abelian group and $G_{T}$ be the set of elements of finite order in $G$ .

Prove that $G_{T}$ is a subgroup of $G$ .
Prove that every non-identity element of $G / G_{T}$ has infinite order.
Characterize the elements of $G_{T}$ when $G = R / Z$ , where $R$ is the additive group of real numbers.

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Let $G$ be an abelian group and $G_T$ be the set of elements of finite order in $G$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that $G_T$ is a subgroup of $G$.
	\item Prove that every non-identity element of $G/G_T$ has infinite order.
	\item Characterize the elements of $G_T$ when $G={\bf R}/{\bf Z}$, where ${\bf R}$ is the additive group of real numbers.
\end{enumerate}