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Dear Laurent, dear Gap Forum,

this is a response Laurent Bartholdi's email to the Forum late last

year. In the mean time we had a exchange of private emails which has

not reached a conclusion yet. This mail describes the status of

existing techniques.

i wondered how one can compute isolators in gap.

recall the isolator of a subgroup H of G is

{x in G | x^n in H for some n}

there is also an interesting series of normal subgroups G_n of G,

where G_n=isolator of gamma_n(G), and {gamma_n} is the LCS.

There are no routines for computing isolators in the standard

distribution of GAP as yet. As part of a joint project for computing

with polycyclic groups, Bettina Eick has written functions that can

compute the torsion subgroup in a nilpotent group given by a

polycyclic presentation. With this it is possible to compute

isolators of the terms in the lower central series. If the group is

given by an arbitrary finite presentation, one can use the nilpotent

nuotient algorithm first to determine a polycyclic presentation for

the factor groups modulo the terms of the lower central series.

also, given a subgroup H of G and an integer n, how does one compute the

subgroup of H generated by nth powers of elements in H?all my groups will be infinite but of finite index in a f.p. group.

they're also residually finite, so it could work (as second best choice)

to work in large enough finite quotients.

I do not know of a general method for computing the group generated by

all n-th powers. Even for the class of finite groups I am not aware

of any efficient method. Computing the subgroup generated by all n-th

powers is equivalent to finding the largest factor group of exponent

n. This question is related to the well-known Burnside problem and in

this context computational techniques exist for prime power exponents

as part of the p-quotient algorithm. Essentially, if the group is

nilpotent, then there is a finite set of n-th powers whose normal

closure is the required normal subgroup. Therefore, it might be

possible to devise a method for this special case.

Werner Nickel.

------------------------------------------------------------------------------ Dr (AUS) Werner Nickel Mathematics with Computer Science Room: 2d/423 Fachbereich Mathematik, AG 2 Tel: +49 6151 163487 TU Darmstadt Fax: +49 6151 166535 Schlossgartenstr. 7 Email: nickel@mathematik.tu-darmstadt.de D-64289 Darmstadt ------------------------------------------------------------------------------

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