> < ^ Date: Thu, 06 Jan 2000 15:18:15 +0100 (CET)
> < ^ From: Werner Nickel <nickel@mathematik.tu-darmstadt.de >
< ^ Subject: Re: powers and roots

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.

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Dr (AUS) Werner Nickel                 Mathematics with Computer Science
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