  A subset $A$ of a group $G$ is called product-free if there is no solution to
$a=bc$ with $a,b,c$ all in $A$. It is easy to see that the largest product-free
subset of the symmetric group $S_n$ is obtained by taking the set of all odd
permutations, i.e. $S_n \setminus A_n$, where $A_n$ is the alternating group.
By contrast, it is a long-standing open problem to find the largest
product-free subset of $A_n$. We solve this problem for large $n$, showing that
the maximum size is achieved by the previously conjectured extremal examples,
namely families of the form $\{\pi~|~\pi(x)\in I, \pi(I)\cap I=\emptyset\}$ and
their inverses. Moreover, we show that the maximum size is only achieved by
these extremal examples, and we have stability: any product-free subset of
$A_n$ of nearly maximum size is structurally close to an extremal example. Our
proof uses a combination of tools from Combinatorics and Non-abelian Fourier
Analysis, including a crucial new ingredient exploiting some recent theory
developed by Filmus, Kindler, Liftshitz and Minzer for global
hypercontractivity on the symmetric group.
