Sign of Permutation is Plus or Minus Unity
Theorem
Let $n \in \N$ be a natural number.
Let $\N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\sequence {x_k}_{k \mathop \in \N_n}$ be a finite sequence in $\R$.
Let $\pi \in S_n$.
Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$.
Let $\map \sgn \pi$ be the sign of $\pi$.
Let $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be defined as:
- $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n} := \map {\Delta_n} {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$
Then either:
- $\pi \cdot \Delta_n = \Delta_n$
or:
- $\pi \cdot \Delta_n = -\Delta_n$
That is:
- $\map \sgn \pi = \begin{cases}
1 & :\pi \cdot \Delta_n = \Delta_n \\ -1 & : \pi \cdot \Delta_n = -\Delta_n \end{cases}$
Thus:
- $\pi \cdot \Delta_n = \map \sgn \pi \Delta_n$
Proof
If $\exists i, j \in \N_n$ such that $x_i = x_j$, then $\map {\Delta_n} {x_1, x_2, \ldots, x_n} = 0$ and the result follows trivially.
So, suppose all the elements $x_k$ are distinct.
Let us use $\Delta_n$ to denote $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$.
Let $1 \le a < b \le n$.
Then $x_a - x_b$ is a divisor of $\Delta_n$.
Then $x_{\map \pi a} - x_{\map \pi b}$ is a factor of $\pi \cdot \Delta_n$.
There are two possibilities for the ordering of $\map \pi a$ and $\map \pi b$:
Either $\map \pi a < \map \pi b$ or $\map \pi a > \map \pi b$.
If the former, then $x_{\map \pi a} - x_{\map \pi b}$ is a factor of $\Delta_n$.
If the latter, then $-\paren {x_{\map \pi a} - x_{\map \pi b} }$ is a factor of $\Delta_n$.
The same applies to all factors of $\Delta_n$.
Thus:
\(\ds \pi \cdot \Delta_n\) | \(=\) | \(\ds \pi \cdot \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pm \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pm \Delta_n\) |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.4$. Kernel and image: Example $142$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 81$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Definition $9.15$