Odd Integers under Addition do not form Subgroup of Integers
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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 $\struct {\Z, +}$
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{E iii}$