Generated Subsemigroup is not necessarily Same as Generated Group

Theorem
Let $\struct {G, \circ}$ be a group.

Let $A \subseteq G$ be a subset of $G$.

Let $\struct {S, \circ}$ be the subsemigroup of $\struct {G, \circ}$ generated by $S$.

Let $\struct {H, \circ}$ be the subgroup of $\struct {G, \circ}$ generated by $S$.

Then it is not necessarily the case that $\struct {S, \circ}$ is the same as $\struct {H, \circ}$.

Proof
Proof by Counterexample:

Let $\struct {\Z, +}$ be the additive group of integers.

Let $A$ be the set of positive odd integers.

From Generator of Subsemigroup: Positive Odd Numbers:
 * the subsemigroup of $\struct {\Z, +}$ generated by $A$ is the semigroup of strictly positive integers under addition

while from Generator of Subgroup: Positive Odd Numbers
 * the subgroup of $\struct {\Z, +}$ generated by $A$ is $\struct {\Z, +}$ itself.