Subgroup generated by shifted set

Question

Let $G$ be a group, $A\subseteq G$ and $a_0$ a fixed element of $A$. It is clear that $\langle a_0A \rangle\leq \langle A \rangle$ and $\langle a_0^{-1}A \rangle\leq \langle A \rangle$.

Is it true that $\langle a_0^{-1}A \rangle\leq \langle a_0A \rangle$ or $\langle a_0A \rangle\leq \langle a_0^{-1}A \rangle$?

Answer 1

For any $a\in A$, $a_0^2$ and $a_0a$ are in $a_0A$, and so $a_0^{-1}a=(a_0^2)^{-1}(a_0a)\in\langle a_0A\rangle$, and so $\langle a_0^{-1}A\rangle\leq\langle a_0A\rangle$.