Action of the Euclidean group, generalizing linearity?

Question

I have a vector $v \in \mathbb{R}^2$ and two elements $(A,a)$ and $(B,b)$ of the Euclidean group $E(2)$. If the relation $$[(A,a)(B,b)](v) = v$$ holds, can I say that $(A,a)(B,b)$ is the neutral element of $E(2)$?

If the elements of the Euclidean group acted as linear transformations, it would be true, but they don't: $(A,a)(v + w) = (A,a)(v) + (A,0)(w) \neq (A,a)(v) + (A,a)(w)$.

Answer 1

No. This isn't even true for linear transformations; take $A$ to be any nontrivial rotation about the origin, $B=I$, and $v=0$. Then $ABv=v$ but $AB\neq I$.