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|+ε fraction of the constraints of a given k-LIN instance over an abelian group, even if there is an assignment that satisfies (1−ε) fraction of the constraints, for any constant ε>0. Engebretsen, Holmerin and Russell (2004) later showed that the same hardness result holds for k-LIN instances over any finite non-abelian group.