## The orbits of corresponding left and right group actions are the same

Suppose $G$ acts on $A$ on the left via $g \cdot a$. Denote the corresponding right action of $G$ on $A$ by $a \cdot g$; that is, $a \cdot g = g^{-1} \cdot a$. Prove that the induced equivalence relations $\sim_\ell$ and $\sim_r$, defined by $a \sim_\ell b$ if and only if $a = g \cdot b$ for some $g \in G$ and $a \sim_r b$ if and only if $a = b \cdot g$ for some $g \in G$, are the same relation. (That is, prove that $a \sim_\ell b$ if and only if $a \sim_r b$.)

If $a \sim_\ell b$, then $a = g \cdot b$ for some $g \in G$. Then $g^{-1} \cdot a = b$, so that $a \cdot g = b$. Thus $b \sim_r a$, and since $\sim_r$ is reflexive, $a \sim_r b$.

If $a \sim_r b$, then $a = b \cdot g$ for some $g \in G$. Then $a \cdot g^{-1} = b$, so that $g \cdot a = b$. Thus $b \sim_\ell a$, and since $\sim_\ell$ is reflexive, $a \sim_\ell b$.