Permutations and Transpositions

[Bashyboy]

Homework Statement

Attached are some screen shots of portion of the textbook I'm currently working through:

Homework Equations

The Attempt at a Solution

My first question, why exactly can't $\Delta$ contains $x_p - x_q$ only once (note, switched from $i,j$ to $p,q$)? As you can see, the author didn't give many very details concerning this. Clearly $\Delta$ can also be written $\Delta = \prod_{(i,j) \in S} (x_i - x_j)$, where $S = \{(i,j) ~|~ 1 \le i < j \le n \}$. Since sets don't contain duplicates of elements, $S$ won't contain any pair $(p,q)$ twice, implying that $x_p - x_q$ won't appear in $\Delta$ more than once. Would this be the reason, that $S$ cannot contain duplicates? Seems to be a rather unremarkable reason, but if it gets job done...

Next, I am trying to prove that $\sigma(\Delta)$ contains either $x_p - x_q$ or $x_q - x_p$, but not both. For simplicity, let $g = \sigma^{-1}$. Suppose that $\sigma (\Delta)$ contains both factors. Then $\sigma (\Delta) = (x_p - x_q)(x_q - x_p) \prod_{(i,j) \in S \setminus \{(p,q),(q,p)\}}$, and therefore

$$g (\sigma(\Delta)) = (x_{g(p))} - x_{g(q)})(x_{g(q)} - x_{g(p)}) \prod (x_{g(i)} - x_{g(j)})$$

$$\Delta = - (x_{g(p))} - x_{g(q)}) (x_{g(p))} - x_{g(q)}) \prod (x_{g(i)} - x_{g(j)})$$,

showing that $\Delta$ contains $(x_{g(p))} - x_{g(q)})$ twice, contradicting what we showed above.

I know: it isn't great. For one thing, the RHS could be $- \Delta$, so that is one flaw in the argument. I hope someone can help. For all DF's verbosity, it doesn't really clearly spell out the details very well, which is why I don't like DF very much, although it has massive number of problems.

 

[haruspex]

why exactly can't Δ contains $x_p - x_q$ only once

I presume you mean, why not more than once.
It's because of the definition of the product. It is taken over the pairs (i,j) for which i<j. Therefore for a given pair of indices with i<j, the pair (i,j) occurs exactly once and the pair(j,i) does not occur at all.

For the same result after permuting the indices, it seems reasonably obvious to me, so I would be happy with the text as it stands. But if you feel it needs to be proved, your proof looks ok.