Permutations Properties: transpositions

Question

Let $\tau_{ij}$ be a transposition if degree n. What does it mean when one says that $\tau_{ij}=\tau_{ji}$? Thanks in advance!

Answer 1

I suppose that what you have in mind is $\tau_{ij}\colon\{1,2,\ldots,n\}\longrightarrow\{1,2,\ldots,n\}$, with $i,j\in\{1,2,\ldots,n\}$ and $i\neq j$, which is the transposition defined by$$\tau_{ij}(k)=\begin{cases}j&\text{ if }k=i\\i&\text{ if }k=j\\k&\text{ otherwise.}\end{cases}\tag1$$Then $\tau_{ij}=\tau_{ji}$ simply means that the functions $\tau_{ij}$ and $\tau_{ji}$ are the same function, which should be obvious from $(1)$.

Answer 2

It means that swapping $i$ and $j$ is the same as swapping $j$ and $i$.