Examples

• The sign of a permutation and its inverse are the same. There are various ways to see this, one of which is based on the multiplicative property of the sign. Since gg^-1 = e, we find
  sgn(g) · sgn(g^-1) = sgn(e) = 1,

  so that sgn(g) and sgn(g^-1) must both be 1 or both be -1.

• Every m-cycle (a₁, ..., a_m) can be written as the product of m - 1 transpositions:
  (a₁, ..., a_m) = (a₁, a₂)(a₂, a₃) ··· (a_{m - 1}, a_m).

  Since transpositions are odd, the multiplicativity of the sign implies that the sign of an m-cycle is (-1)^{m - 1}, i.e., a cycle of even length is odd and a cycle of odd length is even.