  We consider whether given a simple, finite description of a group in the form
of an algorithm, it is possible to algorithmically determine if the
corresponding group has some specified property or not. When there is such an
algorithm, we say the property is \textit{recursively recognizable} within some
class of descriptions. When there is not, we ask how difficult it is to detect
the property in an algorithmic sense.
  We consider descriptions of two sorts: first, recursive presentations in
terms of generators and relators, and second, algorithms for computing the
group operation. For both classes of descriptions, we show that a large class
of natural algebraic properties, \emph{Markov properties}, are not recursively
recognizable, indeed they are $\Pi^0_2$-hard to detect in recursively presented
groups and $\Pi^0_1$-hard to detect in computable groups. These theorems
suffice to give a sharp complexity measure for the detection problem of a
number of typical group properties, for example, being abelian, torsion-free,
orderable. Some properties, like being cyclic, nilpotent, or solvable, are much
harder to detect, and we give sharp characterizations of the corresponding
detection problems from a number of them.
