If $A^2$ is regular, does it follow that $A$ is regular?
My attempt on a proof:
Yes, for contradiction assume that $A$ is not regular. Then $A^2 = A \cdot A$.
Since concatenation of two non-regular language is not regular $A^2$ cannot be regular. This contradicts our assumption. So $A$ is regular. So if $A^2$ is regular then $A$ is regular.
Is the proof correct?
Can we generalize this to $A^3$, $A^4$, etc...? And also if $A^*$ is regular then $A$ need not be regular?
Example: $A=\lbrace 1^{2^i} \mid i \geq 0\rbrace$ is not regular but $A^*$ is regular.