5

I'm a bit confused about permutation groups. I was reviewing for a test and a problem we had was, let τ,o both be in S3. And o(1)=1 and τ(i)=1. Show that if γ ∈ τo$τ^{-1}$ then γ(i) = i.

I started off from right to left, i.e first I evaluated $τ^{-1}$(i) and then took o($τ^{-1}$(i)) and so on. I got the question wrong, and looking at the solutions it makes more sense to go from left to right, i.e to first evaluate τ(i) and then o(1) and so on.

But I thought when you compose permutations you go from left to right? So if you had, A, B as permutations in cycle notation with A=(1,2,3) and B= (1,2) And you took AB wouldn't you first evaluate it as, 1 goes to 2 under B then 2 goes to 3 under A, thus we would have AB=(1,3) with 2 going to itself? When do you evaluate from right to left and vice versa?

Roger Tynes
  • 301

1 Answers1

4

Unfortunately, both conventions are in use: some people evaluate from left to right, some from right to left. You simply have to know which convention is in use. In the exam problem you can actually infer the convention from the question. You need to evaluate $\gamma(i)$; depending on which convention is in force, this means evaluating either $\tau(i)$ first or $\tau^{-1}(i)$ first. The information given in the problem doesn’t tell you what $\tau^{-1}(i)$ is, but it does tell you that $\tau(i)=1$. Thus, you can be pretty sure that the convention in force here is left-to-right composition: perform $\tau$ first, then $o$, and finally $\tau^{-1}$. And of course when you do, everything works out just right: $\tau(i)=1$, $o(1)=1$, and then $\tau^{-1}(1)=i$, so $\gamma(i)=i$.

(By the way, you seem to be reversing left and right: in your $AB$ example you’re actually doing right-to-left evaluation, since you’re applying $B$ first.)

Brian M. Scott
  • 616,228