Sign of a permutation

That the sign is well defined is a theorem: a permutation can be written in many ways as a product of transpositions, but the number of factors always has the same parity. One proof uses the product \(\prod_{i<j}(x_j-x_i)\), which changes sign with each transposition of the variables.

What matters most is that the sign is multiplicative: \(\operatorname{sgn}(\sigma\tau)=\operatorname{sgn}(\sigma)\operatorname{sgn}(\tau)\). In other words, \(\operatorname{sgn}\colon S_n\to\{\pm1\}\) is a homomorphism, and also \(\operatorname{sgn}(\sigma^{-1})=\operatorname{sgn}(\sigma)\). The even permutations form the kernel, the alternating group \(A_n\), which has index 2 and hence \(n!/2\) elements for \(n\ge2\).

To compute it, decompose into disjoint cycles: a cycle of length \(k\) has sign \((-1)^{k-1}\). Thus \((1\ 2)(3\ 4\ 5)\) has sign \((-1)(+1)=-1\). A common mistake is to believe that cycles of even length are even; it is the other way around.