Is there a simple abelian group $G$ with infinite order?

Question

I am reading "An Introduction to Algebraic Systems" by Kazuo Matsuzaka.

There is the following problem in this book:

On p.80
Problem 8:
Show that a simple abelian group $G \neq \{e\}$ is a cyclic group whose order is prime.

Did the author intend the following problem?

If this problem were the following, I could solve it:

Problem 8':
Show that a simple finite abelian group $G \neq \{e\}$ is a cyclic group whose order is prime.

Proof:
Because $G \neq \{e\}$, there is an element $g \in G$ which is not equal to $e$.
$H := \{g^i | i \in \mathbb{Z}\}$ is a subgroup of $G$ and $G$ is abelian.
So $H$ is a normal subgroup of $G$.
And $G$ is simple.
And $H \ni g \ne e$.
So $H = G$.
Let $n := \#H = \#G$.
Then, $n$ is prime.
If $n$ is not prime, then we can write $n = d d'$, $1 < d < n$, $1 < d' < n$.
Then $H' := \{(g^d)^i | i \in \mathbb{Z}\}$ is a subgroup of $G$ whose order is $d'$.
So, $H'$ is neither $\{e\}$ nor $G$.
Becasue $G$ is abelian, $H'$ is a normal subgroup of $G$.
And $G$ is simple.
This is a contradiction.

Answer 1

Suppose that $G$ is simple Abelian, if $x$ in $G$ has an infinite order the group generate by $2x$ is a proper normal subgroup since it does not contain $x$.

If $x$ has a finite order, there exists $y$ not in $H$ the group generated by $x$, and $H$ is normal and proper.

Answer 2

Since $G$ is abelian, all the subgroups are normal. So $G$ simple abelian means there are no proper subgroups.

So take $x\neq e \in G$.

Claim: $\langle x\rangle =G$.

If not, $\langle x\rangle $ is a proper normal subgroup. Contradiction.

So $G$ is cyclic, and contains no proper subgroups.

Claim: $\mid G\mid= p$, where $p$ is prime.

Suppose not. Then either $\mid G\mid=n$, where $n=rs$ for some $r,s\neq1$, or $G$ is infinite. Take $x\neq e$. In the first case, $\langle x^r\rangle \le G$ is proper. Contradiction. In the second case, $\langle x^2\rangle\le G $ is proper (this requires a little argument which I will leave to you). Also a contradiction.