Stephan Rosebrock asked for a function that tests if there is a
homomorphism of a finitely presented group into a given symmetric (or
alternating) group. Jacob Hirbawi explained, how to use the table of
marks of the finitely presented group for this task. This will of
course only work if the finitely presented group is also finite (and
not too big) so that GAP can calculate its table of marks, but then it
is a nice method.

However there is also a method to find permutation representations of
a finitely presented group of a given (not too big) degree (i.e.
homomorhisms into a given symmetric group of not too big degree),
which does not at all assume that the finitely presented group is
finite, namely the so-called Low Index Method which is implemented in
GAP by the function LowIndexSubgroupsFpGroup, described in section
22.6, page 419 of the manual of GAP 3. The practical limitation of the
degree for which the method will work depends strongly on the number
of generators of the finitely presented group but it has proved a
rather useful tool for the investigation of several finitely presented
groups that appeared in the literature. If wanted I can provide some
references.

Joachim Neubueser