Give a one-sentence synopsis of Proof 1 of Theorem 9.15.

Data from 9.15 Theorem

No permutation in S_n can be expressed both as a product of an even number of transpositions and as a product of an odd number of transposition.

Proof 1 of 9.15 Theorem

We remarked in Section 8 that S_A ≈ S_B if A and B have the same cardinality. We work with permutations of then rows of then x n identity matrix I_n, rather than of the numbers 1, 2, ... , n. The identity matrix has determinant 1. Interchanging any two rows of a square matrix changes the sign of the determinant. Let C be a matrix obtained by a permutation a of the rows of In. If C could be obtained from I_n by both an even number and an odd number of transpositions of rows, its determinant would have to be both 1 and -1, which is impossible. Thus a cannot be expressed both as a product of an even number and an odd number of transpositions.