Discrete Algebraic Methods

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By Corollary 1.17, we see that the definition of sgn is independent of the chosen order of elements in {1, . . . , n}, because the definition of a transposition is independent of this order. So, if X is a finite set and π a permutation of X, we can define sgn(π). In particular, considering a finite group G, we can start with any order of G’s elements and every order defines the same sign sgn(g) ∈ {1, −1} for g ∈ G; to see this, consider the permutation x ↦ gx on G induced by g.

Remark 1.19. The kernel of sgn: Sn → {1, −1} is the alternating group An on n elements. These are the permutations with sign 1. The elements of An are also called even permutations, while permutations of sign −1 are called odd permutations. Corollary 1.17 shows that ...