I am struggling with this question for a very long time and just can't find the flaw. So I am given a false Theorem:
The language $\{awwa \mid w \in \{a,b\}^* \}$ is regular.
Well, that part is obvious, we can prove that the language is not regular using the pumping lemma.
The question I am asked is to find the flaw in the "proof" for this theorem:
Let
$\qquad L_1 = \{ aw \mid w \in \{a,b\}^* \}$
$\qquad L_1^R = \{ wa \mid w \in \{a,b\}^* \}$
Let $L_2 = L_1 L_1^R = \{ awwa \mid w \in \{a,b\}^*\}$ be even-length palindromes that begin and end with an $a$. Since $L_1$ is regular, and the class of regular languages is closed under reversal and concatenation, we conclude $L_2$ is also regular.
Can you find the flaw? I could build a DFA for $L_1$ and $L_1^R$, so I know they are regular. And regular languages are closed under reversal and concatenation.
But, however, $L_2$ is still not regular, so where is the mistake in the "proof"?