  It is well known that the triviality problem for finitely presented groups is
unsolvable; we ask the question of whether there exists a general procedure to
produce a non-trivial element from a finite presentation of a non-trivial
group. If not, then this would resolve an open problem by J. Wiegold: `Is every
finitely generated perfect group the normal closure of one element?' We prove a
weakened version of our question: there is no general procedure to pick a
non-trivial generator from a finite presentation of a non-trivial group. We
also show there is neither a general procedure to decompose a finite
presentation of a non-trivial free product into two non-trivial finitely
presented factors, nor one to construct an embedding from one finitely
presented group into another in which it embeds. We apply our results to show
that a construction by Stallings on splitting groups with more than one end can
never be made algorithmic, nor can the process of splitting connect sums of
non-simply connected closed 4-manifolds.
