Submit progress

on Tableaux problem 28

The statement

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 $G$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.