Permutation Composition/Multiplication

Question

Are permutation compositions and multiplications the same thing?

I am confused about when I should be going right to left, and when I should be going left to right when I compute the permutations.

For example, $(x_0, x_1, x_2) \circ (x_0, x_1) = (x_1, x_2)$ or $(x_0, x_2)$?

I'm totally lost on whether I should go left to right or right to left, or whether multiplications and compositions are different/same?

Please enlighten me!

Answer 1

Yes, permutation compositions and multiplications are the same thing.

In your example, let's call $\sigma_1=(x_0,x_1,x_2)$ and $\sigma_2= (x_1,x_2)$.

$\sigma_1∘\sigma_2=(x_0,x_1,x_2)∘(x_0,x_1)=(x_0,x_1,x_2)(x_0,x_1)=(x_1,x_2)$

you go left to right: find $x_0$ in $\sigma_1$, it is sent in $x_1$, now jump to see if in $x_1$ appears in $\sigma_2$. It does, so see who's next to $x_1$ in $\sigma_2$: $x_0$. So $\sigma_1∘\sigma_2$ will not move $x_0$. Don't write anything.

Now start again with the next element in $\sigma_1$: $x_1$. $\sigma_1$ sends $x_1$ in $x_2$ and $\sigma_2$ does not move $x_2$. So $\sigma_1∘\sigma_2$ will send $x_1$ in $x_2$. Write $(x_1,x_2...$

And go on seeing what happens to $x_2$: $\sigma_1$ moves $x_2$ in $x_0$ and $\sigma_2$ moves $x_0$ to $x_1$. So $\sigma_1∘\sigma_2$ will send $x_2$ in $x_1$: you can close the parenthesis $(x_1,x_2)$

It's easier to do than to explain. Do some tries yourself and remember, if you are in doubt, to represent permutations the laborious way:

$\sigma_1=\begin{pmatrix}x_0&x_1&x_2\\x_1&x_2&x_0 \end{pmatrix}$ , $\sigma_2=\begin{pmatrix} x_0&x_1&x_2\\x_1&x_0&x_2\end{pmatrix}$. With this representation it's easy to see that

$\sigma_1\sigma_2=\begin{pmatrix}x_0&x_1&x_2\\x_0&x_2&x_1\end{pmatrix}$. But train to use the cycle representation, it's really faster!

Answer 2

The standard operation on the group of permutations is composition---remember that permutations are just (bijective) functions, so all you're really doing is composing functions. However, it's common to refer to this operation as multiplication, as we typically do in most abstract group settings (the usual exception being when the group is abelian, where we use addition).

So if you're working with permutations, I would always assume the operation is composition---and thus you want to compute permutations right to left---unless otherwise stated.