Odd Integers under Addition do not form Subgroup of Integers

Theorem
Let $S$ denote the set of odd integers.

Then $\struct {S, +}$ is not a subgroup of the additive group of integers $\struct {\Z, +}$.

Proof
Consider the odd integers $1$ and $3$.

We have that $1 + 3 = 4$.

But $4$ is not odd.

Thus addition on $\struct {S, +}$ is not closed.

Hence $\struct {S, +}$ is not a group, let alone a subgroup of $\struct {\Z, +}$