Prove a class of regular languages is not closed under a weird concatenation operation

Question

Let's say we have an operation $L$ and a language $S$. $L(S) = \{s^n ~|~ s \in S, n \geq 0\}$. How can I prove a class of regular languages is not closed under this operation?

Answer 1

Hint. Let $L = ab^*$. Show that $$ S(L) \cap ab^*ab^* = \{ab^nab^n \mid n \geqslant 0 \} $$ and prove that this language is not regular.