I'm recently reading a group theory textbook (Pierre Ramond's book) for physicists and there is a sentence that reads
The permutation group on four objects, generated by the cycles (1 2), (3 4) is intransitive because it does not contain the permutation that maps [1, 2, 3, 4] to [3, 2, 1, 4].
Here, [1, 2, 3, 4] denotes the order of four objects after a group action.
However, I cannot understand it since there is no element which send those four objects to [3, 2, 1, 4] in the first place.
In other words, an intransitive group can also be defined as a group whose the number of orbits is greater than 1.
In this sence, the group generated by (1 2), (3 4) has only one orbit. I think it is trivial since the explicit elements of the group are given by: \begin{align} \{ e, (1 2), (3 4), (1 2)(3 4)\}. \end{align}
Where did I make a mistake? I look forward to your answer/comments.
The given reason is that the group lacks one specific element of the full permutation group $S_4$. Howevre, a better (i.e., correct) reason why this group (or rather its canonical action on the set $\{1,2,3,4\}$) is not transitive is that we can find elements, e.g., $x=1$ and $y=3$, such that there is no element $g\in G$ such that $g(x)=y$.
The two orbits of the action of $G$ on $\{1,2,3,4\}$ are $\{1,2\}$ and $\{3,4\}$.