  Property testing has been a major area of research in computer science in the
last three decades. By property testing we refer to an ensemble of problems,
results and algorithms which enable to deduce global information about some
data by only reading small random parts of it. In recent years, this theory
found its way into group theory, mainly via group stability.
  In this paper, we study the following problem: Devise a randomized algorithm
that given a subgroup $H$ of $G$, decides whether $H$ is the whole group or a
proper subgroup, by checking whether a single (random) element of $G$ is in
$H$. The search for such an algorithm boils down to the following purely group
theoretic problem: For $G$ of rank $k$, find a small as possible test subset
$A\subseteq G$ such that for every proper subgroup $H$, $|H\cap A|\leq
(1-\delta)|A|$ for some absolute constant $\delta>0$, which we call the
detection probability of $A$. It turns out that the search for sets $A$ of size
linear in $k$ and constant detection probability is a non-commutative analogue
of the classical search for families of good error correcting codes. This paper
is devoted to proving that such test subsets exist, which implies good
universal error correcting codes exist -- providing a far reaching
generalization of the classical result of Shannon.
  In addition, we study this problem in certain subclasses of groups -- such as
abelian, nilpotent, and finite solvable groups -- providing different
constructions of test subsets for these subclasses with various qualities.
Finally, this generalized theory of non-commutative error correcting codes
suggests a plethora of interesting problems and research directions.
