Optimal inapproximability of satisfiable k-LIN over non-abelian groups

A seminal result of Håstad (2001) shows that it is NP-hard to find an assignment that satisfies 1/|G|+? fraction of the constraints of a given k-LIN instance over an abelian group, even if there is an assignment that satisfies (1-?) fraction of the constraints, for any constant ?>0. Engebretsen, Holmerin and Russell (2004) later showed that the same hardness result holds for k-LIN instances over any finite non-abelian group. Unlike the abelian case, where we can efficiently find a solution if the instance is satisfiable, in the non-abelian case, it is NP-complete to decide if a given system of linear equations is satisfiable or not, as shown by Goldmann and Russell (1999). Surprisingly, for certain non-abelian groups G, given a satisfiable k-LIN instance over G, one can in fact do better than just outputting a random assignment using a simple but clever algorithm. The approximation factor achieved by this algorithm varies with the underlying group. In this paper, we show that this algorithm is optimal by proving a tight hardness of approximation of satisfiable k-LIN instance over any non-abelian G, assuming P# NP. As a corollary, we also get 3-query probabilistically checkable proofs with perfect completeness over large alphabets with improved soundness. Our proof crucially uses the quasi-random properties of the non-abelian groups defined by Gowers (2008).