> < ^ Date: Fri, 19 Jun 1992 10:24:12 +0200
> < ^ From: Joachim Neubueser <joachim.neubueser@math.rwth-aachen.de >
^ Subject: letter of J. Neil

John Neil asks for a convenient way to convert from a finitely
presented group to a permutation group or some other way to get a
description that can be used in general group functions.

The answer unfortunately has to be that there cannot be a generally
working method, because already the word problem for finitely
presented groups, i.e. the problem to decide if a word in the
generators of a finitely presented group represents the identity of
that group, is algorithmically unsolvable. All methods for the
investigation of finitely presented groups therefore either are in
essence trial-and-error methods such as the Todd Coxeter coset
enumeration method, which *tries* to find a permutation
representation, or only find factor groups of prescribed type like the
p-Nilpotent Quotient algorithm. You can use these in GAP, but you have
to decide whether from some additional information - e.g. you may
know that your group is a p-group or that the group has a subgroup
with the property that the permutation representation on its cosets is
faithful- you are sure that you obtain an isomorphic copy of you given
f.p. group. In GAP however -as it should be done from the theoretical
point of view the f.p. group and its isomorphic image which may e.g.
be a permutation group, are treated as different groups. You may use
the facilities that GAP provides for setting up and using
homomorphisms to establish e.g. an isomorphism between the f.p. group
and its faithful image in a permutation representation.

Joachim Neubueser

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