My book (Bishop, David m. Group Theory and Chemistry) states, that $C_3C_3^{-1} = E$ and $C_3^{-1}C_3 = E$.
After that it shows an non-commutative example:
Ok. But then they say that $(PQ)R = P(QR) = PQR$. Actually, the operations are always applied from right to left, but here the sequence $Q, P, R$ and $R, Q, P$ is apparently the same. This is what confuses me a little, although it was shown before that it generally depends on the sequence of the linkage.
No.
Consider the rotations and reflections of an equilateral triangle. It is not difficult to see that, if we label the vertices of the triangle by $1,2,$ and $3$, then the symmetries correspond to the group $S_3$ of permutations of three elements. (In fact, this can be made precise: the group $D_3$ of symmetries of an equilateral triangle is isomorphic to $S_3$.)
However, $(12)(23)=(312)$, whereas $(23)(12)=(132)\neq (312)$.
Thus $D_3$ is not abelian.
But all groups are associative, which is to say that for all $a,b,c\in G$, where $G$ is a group, we have
$$a(bc)=(ab)c.$$