> < ^ Date: Wed, 14 Jul 1993 15:33:02 +0200
> < ^ From: Joachim Neubueser <joachim.neubueser@math.rwth-aachen.de >
< ^ Subject: Re: DimensionLoewyFactors

Thierry Dana-Picard asked:
> I entered some small p-groups as finitely presented groups and used
> the function DimensionsLoewyFactors. For the first ones everything
> seemed OK, but once I got the following:
> gap> g:=Group(a,b,c);
> Group( a, b, c )
> gap> g.name:="g";
> "g"
> gap> g.relators:=[a^3,b^3,c^3,a*b*a^2*b^2,a*c*b*a^2*c^2,b*c*b^2*c^2];
> [ a^3, b^3, c^3, a*b*a^2*b^2, a*c*b*a^2*c^2, b*c*b^2*c^2 ]
> gap> Size(g);
> 27
> gap> DimensionsLoewyFactors(g);
> Error, This function will not work for finitely presented groups.
> Please use 'OperationCosetsFpGroup' in order to construct
> an isomorphic permutation group in
> G.operations.ConjugacyClasses( G ) called from
> ConjugacyClasses( G ) called from
> G.operations.Agemo( G, p ) called from
> AgemoAbove( L[i], C, p ) called from
> G.operations.JenningsSeries( G ) called from
> ..
> brk>
> What happens here?
> Furthermore, I got the same
> > Please use 'OperationCosetsFpGroup' in order to construct
> > an isomorphic permutation group
> when trying to compute Centre or ConjugacyClasses of a finitely
> presented group.
> Could somebody send more information and/or examples?

The answer is fairly simple: There are no general algorithms to find
for a finitely presented group its conjugacy classes, centre or
Jennings series (from which in turn the dimensions of the factors of
the Loewy series are obtained). This is a consequence of the
unsolvability of the word problem. However, if the finitely presented
group is indeed finite, a faithful permutation representation can be
found and used. GAP does not try to do this automatically but asks the
user to use 'OperationCosetsFpGroup' to do just that.

Just in very special cases, e.g. if the presentation exhibits that the
group is in fact finite abelian, the function 'DimensionsLoewyFactors'
will recognise this and directly give the answer. This is perhaps a
slightly misleading act of courtesy since it is then in fact puzzling
why a function that did answer for some very small examples suddenly
does not work.

By the way, if you want to compute the dimensions of the Loewy factors
of a p-group it is advisable to describe the group by an AG
(polycyclic) presentation from which GAP can work directly.

I hope that this clarifies the situation. Joachim Neubueser

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