Exercise 41, § 6

Let $x, y$ be two elements of a group $G$.

(a) For there to exist $a, b$ in $G$ such that $bay = xab$, it is necessary and sufficient that $xy^{-1}$ be a commutator.

(b) For there to exist $2n + 1$ elements $a_1, a_2, \ldots, a_{2n+1}$ in $G$ such that

it is necessary and sufficient that $xy^{-1}$ be a product of $n$ commutators (argue by induction on $n$).