Calculating products of permutations

Question

This is a very simple question, but if I want to compute say $(12)(13)$, I see this as $(12)(13)[1,2,3] = (12)[3,1,2]$ which flips the elements 1 and 3. Now to calculate $(12)[3,1,2]$, do we flip the elements 1 and 2, or the elements in positions 1 and 2 (these would give us [3,2,1] and [1,2,3] respectively)?

Answer 1

A permutation is a function from $\{1,2,\ldots,n\}$ to itself. Multiplication of permutations is function composition: since we write functions on the left, we usually define $$(\sigma\tau)(k)=\sigma(\tau(k)).$$ For $\sigma=(1\,2)$ and $\tau=(1\,3)$ then $$\sigma\tau(1)=\sigma(3)=3,$$ $$\sigma\tau(2)=\sigma(2)=1$$ and $$\sigma\tau(3)=\sigma(1)=2.$$ Therefore $(1\,2)(1\,3)=(1\,3\,2)$.