In a question for an assignment (specific group but I've abstracted the question for academic integrity). I was given a group $G=\langle\ a,b \mid R\ \rangle$ where $R$ is some set of relations. Then, the question defines a group action on $X$ for each generating element. The first question is then "Verify that this gives a group action on $X$".
My problem is that it seems like by defining a group action on generators inherently assumes that $e(x)=x$ and $a(b(x))=ab(x)$ from the beginning, so it seems like there is nothing left to show?
Since the question only defined the functions $a(x)=?$ and $b(c)=?$, I can't verify that $ab(x)=a(b(x))$, since I wasn't given the action defined by $ab$.
Is this a misunderstanding on my part, or do you think it is likely that the question itself is ambiguous?