  We analyze a Markov chain, known as the product replacement chain, on the set
of generating $n$-tuples of a fixed finite group $G$. We show that as $n
\rightarrow \infty$, the total-variation mixing time of the chain has a cutoff
at time $\frac{3}{2} n \log n$ with window of order $n$. This generalizes a
result of Ben-Hamou and Peres (who established the result for $G =
\mathbb{Z}/2$) and confirms a conjecture of Diaconis and Saloff-Coste that for
an arbitrary but fixed finite group, the mixing time of the product replacement
chain is $O(n \log n)$.
