Hello I am countering problem when I tried to prove $\epsilon$ is well defined function.
My try: I defined X be set of two element subsets of {1,2,....,n}. For each f in Sn and every { i, j } in X, I defined f'({ i, j }) =$\frac{ f(j)-f(i)}{(j-i)} $ . If I show f' is well defined then f will be well defined. I have chosen conventional way to show f' to be well defined. Let { i, j } and { l,m } be in X and f'({ i, j })=f'({ l, m }). From now I am stuck and don't know how to proceed further. Please give me a hint or something. Thank you
Let $n=3$ and let $\sigma\in S_3$ be such that $$\begin{align} \sigma(1)&=2\\ \sigma(2)&=3\\ \sigma(3)&=1 \end{align}$$ Then $$\epsilon(\sigma)=\prod_{1\le i<j\le 3}{\sigma(j)-\sigma(i)\over j-i}= {\sigma(2)-\sigma(1)\over2-1}{\sigma(3)-\sigma(1)\over3-1}{\sigma(3)-\sigma(2)\over3-2}=\\={3-2\over2-1}{1-2\over3-1}{1-3\over3-2}=1$$
We can perform a similar calulation for any $\sigma\in S_n$ for any $n.$ There is no question that $\epsilon$ is well-defined. The only choice that is made is the order in which to list the factors in the product, and of course, this doesn't matter.