Question: does there exist an infinite abelian group having exactly $n$ elements of finite order?
My attempt: yes! consider the most familiar, infinite abelian group $\mathbb{Z}$ (of course with respect to $+$) and consider the additive group of integers modulo $n$ that is, $\mathbb{Z_n}$. Now consider, $G=\mathbb{Z} ×\mathbb{Z_n}$, then clearly, $G$ is infinite abelian group (since both $\mathbb{Z}$ and $\mathbb{Z_n}$ are abelian).
Further, as $\mathbb{Z}$ has exactly one element of finite order (which is identity $0$) it follows that the group $G$ has exactly $n$ elements of finite order and they are, $(0,0), (0,1),...,(0,n-1)$. Am I correct?
'Can we have more interesting examples like this'?