how to convert right group action to left group action?

Question

In Wikipedia it says one can convert right group action to left group action, because of the formula $(gh)^{−1} = h^{−1}g^{−1}$.
Can you explain how this works?

Answer 1

If you have a left action you want to make into a right action, say $G$ acts on $X$, then just define $x \cdot g := g^{-1} \cdot x$ for $g \in G$ and $x \in X$. The formula you've quoted above is then exactly what you need to prove that this gives a group action. (If you did this without the inverse in there, then you wouldn't get a group action -- check this!)