Permutations: Interpreting Image Notation I have a problem in interpreting permutation. I think the definition and my interpretation of it don't match each other.
Let $\sigma=(1\ 2\ 4\ 3)$, and $\tau=(1\ 3\ 2\ 4)$ in one-line notation. 
I thought $\sigma$ is a '3, 4 swapping rearrangement of 1 2 3 4'
and $\tau$ is a '2, 3 swapping rearrangement of 1 2 3 4' like swapping data in a computer memory.
The problem is this : I interpreted $\sigma\tau$ as 'first swap 2 and 3, then you get 1 3 2 4, and then swap the elements in the 3rd and 4th place obtaining 1 3 4 2'.
But the definition says $\sigma\tau=(1\ 4\ 2\ 3)$. I thought in reality, the permutation is just a place-changing operation, but by definition, composing two permutation is not just changing place twice. What is wrong in my interpretation?
 A: When we write $(1243)$, we mean the permutation defined where $1 \mapsto 2$ and $2 \mapsto 4$ and $4 \mapsto 3$ and $3 \mapsto 1$.
Likewise, $(1324)$ means we have $1 \mapsto 3$ and $3 \mapsto 2$ and $2 \mapsto 4$ and $4 \mapsto 1$.  Finally, any number not written inside a set of parentheses with other numbers is presumed to be fixed.  So, for example, if we think of these as permutations in $S_6$, both $\sigma$ and $\tau$ fix $5$ and $6$.  Notice that these are indeed bijective maps!  So the way a layman thinks of the word "permutation" is indeed a correct way of thinking about what's going on.  This notation is called "cycle notation" which kind of makes sense: the permutation can be visualized as equidistant points on a circle, and the circle gets rotated so that each point gets moved to its immediate counterclockwise neighbor.
Now we can compose these with each other.  Doing so, let's see where $1$ is sent.  $\sigma \tau(1) = \sigma ( \tau (1)) = \sigma(3) = 1$.  Looking at $2$ next, we have $\sigma( \tau (2)) = \sigma(4) = 3$.  And so forth.
A: There are multiple levels of confusion, here (and I have deleted my previous answer because of this).
Many posters here are accustomed to cycle notation, where $\sigma = (1\ 2\ 4\ 3)$ means this:
$\sigma(1) = 2\\ \sigma(2) = 4\\ \sigma(3) = 1\\ \sigma(4) = 3.$
It is also possible to employ "image notation" where $\sigma = (1\ 2\ 4\ 3)$ means this:
$\sigma(1) = 1\\ \sigma(2) = 2\\ \sigma(3) = 4\\ \sigma(4) = 3.$
Finally, sometimes $\sigma\tau$ means $\sigma$ first, then $\tau$, and sometimes it means $\tau$ first, then $\sigma$ (as in ordinary function composition).
So it's not immediately clear which conventions your text is using.
A third source of confusion is what $(1\ 2\ 4\ 3)$ is instructing us to do: are we swapping the NUMBERS $4$ and $3$, or whatever numbers are in the $3$rd and $4$th positions? This makes the whole situation even murkier.
It appears your text is using "image notation", and "normal composition", so that:
$\sigma\tau(1) = \sigma(\tau(1)) = \sigma(1) = 1$
$\sigma\tau(2) = \sigma(\tau(2)) = \sigma(3) = 4$
$\sigma\tau(3) = \sigma(\tau(3)) = \sigma(2) = 2$
$\sigma\tau(4) = \sigma(\tau(4)) = \sigma(4) = 3$
which is indeed $(1\ 4\ 2\ 3)$ in "image notation".
