Tableaux problem 28

28.[Stan1999a]*Ch. 7 Exer. 68(b) Let $G$ be a finite group of order $g$. Given $w\in G$, let $f(w)$ be the number of pairs $(u,v)\in G\times G$ satisfying $w=uvu^{-1}v^{-1}$ (the commutator of $u$ and $v$). Thus $f$ is a class function on $G$ and hence a linear combination $\sum c_{\chi}\chi$ of irreducible characters $\chi$ of $G$. The multiplicity $c_{\chi}$ of $\chi$ in $f$ is equal to $g/\chi(1)$. Since $\chi(1)|g$, it follows that $f$ is a character of $G$. Find an explicit $G$-module $M$ whose character is $f$, which directly proves that $f$ is a real character. Since in the equation $f=\sum\frac{g}{\chi(1)}\chi$ all the $\chi$’s are linearly independent, this would provide a new proof of the basic result that $\chi(1)$ divides the order of $G$.