On chapter 2 of Wu-Ki Tung's Group Theory in Physics, he writes:
But a cyclic permutation $(321)$, for example, implies sending 3 to 2, 2 to 1 and 1 to 3. But by doing so, looking at the following triangle:
what we get is a counter-clockwise rotation about the center by $\frac{4\pi}{3}$ for $(321)$, instead of $\frac{2\pi}{3}$. What am I doing wrong?
I decided to give this answer so that everyone that is beginning group theory can grasp through this initial part more quickly.
The cycle notation for the permutation mentioned, (321), comes from decomposing in cycles the following permutation $P$ (that I consider to be acting on the set (1,2,3)) : \begin{equation} P = \begin{pmatrix} 1 & 2 & 3\\ 3 & 1 & 2 \end{pmatrix} \end{equation} Here, the elements in the first row are replaced by the respective elements in the second row so that 1 is replaced by 3, 2 is replaced by 1 and 3 is replaced by 2. This can be written in cycle notation as (132). To make this notation more obvious and to notice that (213) and (321) also refer to the same permutation P, imagine arrows pointing to the right between the elements of each cycle notation. The arrows mean replacing the element before the arrow by the element that the arrow points to. Doing so it's easy to realize that the replacments being perfomed are the same (knowing that the element on the far left replaces the element on the far right):
(1$\rightarrow$3$\rightarrow$2) = (2$\rightarrow$1$\rightarrow$3) = (3$\rightarrow$2$\rightarrow$1).
Thus, this means that the set (1,2,3) goes to (3,1,2) after performing the replacements.
Lets now consider the triangle in Wu-Ki Tung book unchanged (with only the labels that will matter to visualize the rotation). In the image linked below, this triangle is followed on the right by the resulting triangle of a rotation by 120º counter clock-wise and then by the results of another rotation by 120º counter clock-wise (thus a rotation by 240º of the unchanged triangle). Three triangles
It is straightforwad to see that the permutations mentioned in the book (321) and (123) represent the same triangle that was, respectively, rotated by 120º and 240º. What must not be confused is that these cycle notations for permutations really mean a label replacement not a label deviation to the next vertex!
The rotations and reflections that leave the triangle unchanged are the operations that constitute the dihedral group, $D_3$. Only by labeling the vertices of the triangle we can follow what each operation did. But, when we label the vertices we realize that there are now other ways of leaving the triangle unchanged: alterig just the labels does in fact nothing to the triangle. Thus, such labels permutations constitute $S_3$, the symmetric group. Both groups are isomorphic to one another! This means that there is a one to one correspondence between the elements of these groups (a bijection) and also, that the $group$ $operation$ defining each group is preserved (a homeomorphism). We can notice this by realizing that the group multiplication tables are similar.