On the largest product-free subsets of the alternating groups

Abstract

AbstractA subset $A$ A of a group $G$ G is called product-free if there is no solution to $a=bc$ a = b c with $a,b,c$ a , b , c all in $A$ A . It is easy to see that the largest product-free subset of the symmetric group $S_{n}$ S n is obtained by taking the set of all odd permutations, i.e. $S_{n} \backslash A_{n}$ S n ∖ A n , where $A_{n}$ A n is the alternating group. In 1985 Babai and Sós (Eur. J. Comb. 6(2):101–114, 1985) conjectured that the group $A_{n}$ A n also contains a product-free set of constant density. This conjecture was refuted by Gowers (whose result was subsequently improved by Eberhard), still leaving the long-standing problem of determining the largest product-free subset of $A_{n}$ A n wide open. We solve this problem for large $n$ n , showing that the maximum size is achieved by the previously conjectured extremal examples, namely families of the form $\left \{ \pi :\,\pi (x)\in I, \pi (I)\cap I=\varnothing \right \} $ { π : π ( x ) ∈ I , π ( I ) ∩ I = ∅ } 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 $A_{n}$ 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, Lifshitz and Minzer for global hypercontractivity on the symmetric group.