In how many ways can a group element in a finite group be written as a commutator? It seems there is a result by Frobenius that states that the number of ways an element $g$ of a finite group can be written as a commutator ($\phi(g) =  | \{(x,y) \in G \times G: g = [x,y]\}|$) is given by $\phi(g) = \sum_{\chi} \frac{|G| \chi(g)}{\chi(1)}$, where the sum is taken over all the irreducible characters of $G$.
I can't find the original paper and am having trouble on proving this. I'm trying to make use of the class algebra constants, but it's of no use so far. Would anybody kindly provide some advice?
Thank you!
 A: Uber Gruppencharaktere. Sitzungsber. der Berl. Ak., 1896, Seite 985-1021. (See Sect.3 in it).
A: More general results are proved in Alon Amit and Uzi Vishne, Characters and solutions to equations in finite groups, J Alg Appl 10 (2011) 675-686. For the result of Frobenius, they refer to pages 1 to 37 of his Gesammelte Abhandlungen, Band III. They also cite A M A Alghandi and F G Russo, A generalization of the probability that the commutator of two group elements is equal to a given element, arXiv:1004.0943, and T Tambour, The number of solutions of some equations in finite groups and a new proof of Ito's theorem, Commun Alg 28 (2000) 5353-5362. I haven't looked at any of these. 
Another paper that might interest you is M R Pournaki and R Sobhani, Probability that the commutator of two group elements is equal to a given element, J Pure Appl Alg 212 (2008) 727-734. But this paper just cites the result of Frobenius, and refers to the collected works, without page numbers. 
A: Thank you for all your help, but I've managed to find out in how many ways an arbitrary group element can be written as a given word (and of course this proves this topic's statement). I had opened this Number of ways a group element of a finite group can be written as a given word for this more general question.
A: Actually, the solution follows easily if one combines the result of Problem 3.23 in Isaacs' book on characters with the orthogonality relation (once one replaces $h$ by $g^-1$ in that problem). 
To my mind, a different question related to the above is interesting: to get a character-free formula for that number of ordered pairs $(x,y)$ giving $g=[x,y]$ for a fixed $g$ in $G$.
Hope this helps,
Nea Marin
