Generated Subsemigroup is not necessarily Same as Generated Group
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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
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.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings