Let $G$ be a group and $S \subseteq G$. We define the normalizer of $S$ as $N(S) := \{ n \in G : nS = Sn\}$
According to Wikipedia;
If $S$ is a subsemigroup of $G$, then $N(S)$ contains $S$.
But I can't prove it. Everything that $S$ being a subsemigroup tells me is that $\forall s \in S,\ sS \subseteq S$ and $Ss \subseteq S$
How would one derive that $\forall s \in S,\ sS = Ss$ ? If S were a subgroup or finite, it would be trivial.
This is not true. Consider the free group generated by two symbols $a,b$. Let $A$ be its subsemigroup generated by $a,b$. Now $a\in A$, but $aA$ consists of all strings in $a,b$ starting with $a$, while $Aa$ consists of all strings in $a,b$ terminating with $a$. They are obviously different.