Finite Order Elements of Infinite Abelian Group form Normal Subgroup

Theorem
Let $G$ be an infinite abelian group.

Let $H \subseteq G$ be the subset of $G$ defined as:
 * $H := \set {x \in G: x \text { is of finite order in } G}$

Then $H$ forms a normal subgroup of $G$.

Proof
Let $e$ be the identity element of $G$.

From Identity is Only Group Element of Order 1, $\order e = 1$ and so $H \ne \O$.

Let $a \in H$.

Then by Order of Group Element equals Order of Inverse:
 * $\order a = \order {a^{-1} }$

and so $a \in H$.

Let $a, b \in H$.

From Product of Orders of Abelian Group Elements Divides LCM of Order of Product:
 * $\order {a b} \divides \lcm \set {\order a, \order b}$

where:
 * $\order a$ denotes the order of $a$
 * $\divides$ denotes divisibility
 * $\lcm$ denotes the lowest common multiple.

Thus $a b$ is also of finite order.

Thus by definition:
 * $a b \in H$.

By the Two-Step Subgroup Test it follows that $H$ is a subgroup of $G$.

By Subgroup of Abelian Group is Normal, $H$ is normal in $G$.