Some years ago, Keith Dennis posted a nice discussion of signs of permutations on the Group-Pub-Forum mailing list, following some papers by Cartier. I gave it to my graduate algebra class as a series of exercises, which I'll copy below. Dennis also pointed out that Cartier used similar ideas to give alternate viewpoints on the group-theoretic transfer and on Legendre symbols. References to Cartier's papers follow.
The complete graph $K_n$ on $\{1, 2, 3, \ldots, n\}$$\{1, 2, 3, \dotsc, n\}$ is the graph with $n$ vertices and with an edge connecting every pair of distinct vertices. An orientation of $K_n$ is a choice of direction for each edge (formally, the edges are sets consisting of two distinct vertices, and an orientation just selects an ordering for each such set).
Note that we don't require any kind of compatibility between the orderings of different edges.
If $o$ and $o'$ are orientations of $K_n$, we define $m(o, o')$ to be the number of edges on which the orientations differ, and we set $d(o,o') = (-1)^{m(o,o')}$.
Prove the following statements:
a) $d(o,o')d(o',o'') = d(o,o'')$ for all orientations $o, o', o''$ of $K_n$;
b) The symmetric group $S_n$ acts on the set of orientations of $K_n$. Formally, if $o$ is an orientation in which the edge between $i$ and $j$ points from $i$ to $j$, then in the orientation $\sigma \cdot o$, the edge between $\sigma(i)$ and $\sigma(j)$ points from $\sigma(i)$ to $\sigma(j)$. Prove that this defines an action, and show that $d(o, \sigma o) = d(o', \sigma \cdot o')$$d(o, \sigma\cdot o) = d(o', \sigma \cdot o')$ for all orientations $o, o'$ of $K_n$ and all $\sigma \in S_n$.
We can now define $sgn (\sigma) = d(o, \sigma o)$$\operatorname{sgn} (\sigma) = d(o, \sigma o)$.
c) Prove that $sgn: S_n \rightarrow \mathbb Z /2\mathbb Z$$\operatorname{sgn}: S_n \rightarrow \{\pm1\}$ is a homomorphism.
d) Prove that $sgn (\tau) = -1$$\operatorname{sgn} (\tau) = -1$ for every transposition $\tau$.
References:
Zbl 0195.03101 Cartier, P. Remarques sur la signature d'une permutationRemarques sur la signature d'une permutation (French) Enseign. Math., II. Sér. 16, 7-19 (1970).
Zbl 0195.04001 Cartier, P. Sur une géneralisation du transfert en théorie des groupesSur une généralisation du transfert en théorie des groupes (French) Enseign. Math., II. Sér. 16, 49-57 (1970).
Zbl 0195.05802 Cartier, P. Sur une généralisation des symboles de Legendre-Jacobi Sur une généralisation des symboles de Legendre–Jacobi (French) Enseign. Math., II. Sér. 16, 31-48 (1970).
