# Concatenation of two regular languages

In a question, it is given that L is a finite language over the unary alphabet and L+ is not regular. We know that L+ = LL* Since L is finite, it must be regular because all finite languages are regular. If L is regular L* is also regular. Then how can be L+ not regular? It is the concatenation of two regular languages. Where am I wrong?

That's impossible. Since $L$ is finite, it is regular. Since $L$ is regular, $L^+ = L L^*$ is regular. See, e.g., https://en.wikipedia.org/wiki/Regular_language#Closure_properties.