Let $G$ be solvable. Let $g\in G$. What can we say about $g=[g^{x_1}, g^{y_1}]\dots [g^{x_n}, g^{y_n}]$ with $x_i,y_i\in G,i=1,...,n$?

Question

We have $G$ a solvable group. Let g be an element of G. What can we say about

$$g = [g^{x_{1}}, g^{y_{1}}]\cdot\cdot\cdot[g^{x_{n}}, g^{y_{n}}]$$

with $x_{i}, y_{i} \in G, i = 1,..., n$?

Answer 1

If $G$ is finite, then by an absurd-sounding result of John Wilson, $n\leq 55$. This appears in J. London Math. Soc., entitled Finite Axiomatization of Finite Soluble Groups. The theorem is as follows:

A finite group $G$ is soluble if and only if no non-trivial element $g$ can be expressed as a product of 56 commutators of conjugates of $g$.

It is likely that the number 56 can be reduced.