Regarding the composition of permutations...

Question

So I have the permutations: $$\pi=\left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2 & 3 & 7 & 1 & 6 & 5 & 4 & 9 & 8 \end{array} \right)$$ $$\sigma=\left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 9 & 5 & 6 & 8 & 7 & 1 & 2 & 4 & 3 \end{array} \right)$$

I found $\pi\sigma$ and $\sigma\pi$ to be

$$\pi\sigma=\left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 5 & 6 & 2 & 9 & 1 & 7 & 8 & 3 & 4 \end{array} \right)$$

$$\sigma\pi=\left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 8 & 6 & 5 & 9 & 4 & 2 & 3 & 1 & 7 \end{array} \right)$$.

So my question is: did I do these correctly or did I mix them up (i.e, the permutation matrix I have for $\pi\sigma$ is actually $\sigma\pi$)?

Answer 1

You mixed them up.If you look at the permutation matrix $\pi \sigma $ you have to look ,firstly at $\sigma$ and then at $\pi$.

For example,for the first column of $\pi \sigma$,it is:

$$1 \to 9, 9 \to 8$$

Answer 2

You've done just fine - EXCEPT: the first is $\sigma \pi$ and the second is $\pi\sigma$.

$$\sigma \pi = \sigma\circ \pi = \sigma(\pi(i))\quad i = 1, 2, \ldots, 9\, $$ So we can only find $\sigma(\pi(i))$ after finding $\pi(i)$.

Answer 3

It depends.

Where I did my undergrad we learned to write permutations on the right, so $3\sigma\pi=\ldots$. This can be found in older books. The more common (people will claim, "standard") notation today is to write $\sigma\pi(3)$. Therefore, it depends on what your book or lecturer is telling you. If you have been taught to write permutations on the right, then you are correct (yay!), but if you have been taught to write them on the left then you are incorrect.

My personal preference is to write them on the right. This is for a variety of reasons (I work with automorphism groups, and to me writing maps on the right makes seeing how the automorphisms compose much clearer, as $x\phi\psi=(x\phi)\psi$ while (because I want my diagrams to be nice) writing on the right yields $\phi\circ\psi(x)=\psi(\phi(x))$), but the most relevant one here is the following:

When you write permutations in disjoint cycle form, then I find it much clearer for the following reason:

• Writing your maps on the left (common) means you have to read left-to-right in each permutation, but right-to-left through the permutations. For example, when working out where $(42)(341)(2314)$ sends $2$, I see that $2\mapsto 3\mapsto4\mapsto2$. But my brain is mush.
• Writing your maps on the right (less common) means you have to read left-to-right throughout the calculation. For example, now when working out where $(42)(341)(2314)$ sends $2$, I see that $2\mapsto 4\mapsto1\mapsto4$. And my brain is no longer mush!

Having spouted my opinion I should, however, give you a warning: You should follow the standards of your peers and superiors. If your lecturer is telling you to write them on the right, then you do that. If they say "on the left!", then do that. Don't try to force your opinions on them - it is a bad reason to fail an exam! (Or, in my case, have a paper rejected (okay, not the only reason, but it is the one I took away from it...) - and seriously, it is an pain in the arse having to change your notation consistently throughout!).