Suppose that you define a left group action $g: X\rightarrow X$ by $x \in X \mapsto gx \in X$, and a right group action on $X$, too. Do these two actions have to commute? That is, does it have to be that $g(xg') = (gx) g'$ for all $x \in X$ and $g, g' \in G$?
Do left group action and right group action have to commute
group-actionsgroup-theory
Best Answer
Take any noncommutative group $(G, *)$ and define the actions of $G$ on $X = G$ as: $gx = g * x, xg = g^{-1} * x$. Then $g(xg') = g * (g')^{-1} * x$ and $(gx)g' = (g')^{-1} * g * x$, which need not coincide.