Subgroup of Elements whose Order Divides Integer

Lemma
Let $$A$$ be an abelian group.

Let $$B$$ be a set of the form:
 * $$K = \left\{{x \in A : a^k = e}\right\}$$

where $$k \in \Z$$.

Then $$B$$ is a subgroup of $$A$$.

Proof
First note that the identity $$e$$ satisfies $$e^k = e$$ and so $$B$$ is non-empty.

Now assume that $$x, y \in B$$.

Then:

$$ $$ $$ $$

Hence, by the One-Step Subgroup Test, $$B$$ is a subgroup of $$A$$.