I've been struggling with the following question:
Let $L$ be a regular language. We define $OPP(L)$ to be: $OPP(L)=\{uv~|~vu~\text{belong to}~L\}$
prove that if $L$ belongs to $Ldfa$ then $OPP(L)$ also belongs to $Ldfa$.
Thing is, We've learned that $Lreg=Lfa$, meaning, if I was able to prove that $OPP(L)$ is regular, that would mean that it's possible to generate a Finite Automaton for the language.
I tried proving it's regularity using Induction over $|r|$ where $r$ is the regular expression of $L$ (we know $L$ is regular, therefore, a regular expression $r$ exists that $L(r)=L$), but I got kinda lost trying to perform the induction step.
Any help/leads will be much appreciated.