True/False Question :
Let $G$ be an abelian group, with identity element $e$. If
$$\left \{ g \in G \mid g = e\:or\:g\:has\:infinite\:order \right \}$$
is a subgroup of $G$, then either all elements of $G\setminus \left \{ e \right \}$ have infinite order, or all elements of $G$ have infinite order.