MathNet 2019-part-05 — 184 problems (EN–VI) Question 66

Let $G$ be a finite group, and let $x_1, \dots, x_n$ be a labeling of its elements. Consider the $n \times n$ matrix $(a_{ij})$, where $a_{ij} = 1$ if $x_i x_j^{-1} \neq x_j x_i^{-1}$, and $a_{ij} = 0$ otherwise. Establish the parity of the integer $\det(a_{ij})$.
Amer. Math. Monthly