Group action defined on generators. 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?

What you need to check is that every relator will act (by concatenating the action for the generators as given in the relator) as the identity.

An example to show that this is needed would be a cyclic group $\langle g \rangle$ of order $3$, which acts on $4$ points via $g\mapsto (1,2,3,4)$.

Then $g^3$, the identity, would have to act as $(1,4,3,2)$, which is forbidden.