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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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