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.

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