On the number of solutions of a generalized commutator equation in finite groups (Q1714990)

The following result is proved. Theorem. Let $G$ be a finite group and $w(x_{1},\dots,x_{m}):=$ $\left[w_{1}(x_{1},\dots,x_{n}),w_{2}(x_{n+1},\dots,x_{m})\right] $ a non-trivial reduced word. If $\zeta_{G}^{w_{1}}$ is a character of $G$ and $\zeta_{G}^{w_{2}}$ is a constant map then $\zeta_{G}^{w}$ is a character of $G$ and $\zeta_{G}^{w}=\sum_{\chi \in \mathrm{Irr}(G)}\frac{\left\vert G\right\vert ^{m-n}}{\chi (1)}\left\langle \zeta_{G}^{w_{1}}\chi ,\chi \right\rangle \chi $.\par This extends a classical result of Frobenius with $w(x_{1},x_{2})=[x_{1},x_{2}]:=x_{1}^{-1}x_{2}^{-1}x_{1}x_{2}$ and some other results in the area.\par For the word $w(x_{1},\dots,x_{n})=[\dots[[[x_{1},x_{2}],x_{3}],x_{4}],\dots,x_{n}]$, using the theorem, exact formulas for $\zeta_{G}^{w}$ in case of nilpotent Camina groups and its generalizations are obtained.