  A seminal result of H\r{a}stad [J. ACM, 48(4):798--859, 2001] shows that it
is NP-hard to find an assignment that satisfies $\frac{1}{|G|}+\varepsilon$
fraction of the constraints of a given $k$-LIN instance over an abelian group,
even if there is an assignment that satisfies $(1-\varepsilon)$ fraction of the
constraints, for any constant $\varepsilon>0$. Engebretsen et al. [Theoretical
Computer Science, 312(1):17--45, 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 [Information and Computation, 178(1):253--262. 2002].
  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 {\em optimal} by proving a tight hardness of
approximation of satisfiable $k$-LIN instance over {\em any} non-abelian $G$,
assuming $P \neq NP$.
  As a corollary, we also get $3$-query probabilistically checkable proofs with
perfect completeness over large alphabets with improved soundness.
