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.