> < ^ Date: Fri, 22 Jul 1994 09:29:00 +0100 (WET)
> < ^ From: Werner Nickel <nickel@mathematik.tu-darmstadt.de >
^ Subject: Re: space groups and a bug

Dear Forum,

Franz G"ahler reported in the GAP-Forum about a bug he observed with
the command LowIndexSubgroupsFpGroup() and, moreover, raised some
general questions about handling space groups in GAP. Let us answer
these in turn.

1. There is indeed a bug in LowIndexSubgroupsFpGroup(). However, it is
easily corrected. What happens is the following. Let as a second
argument of that command a non-trivial subgroup U be given and
therefore, subgroups V containing U be sought. Then the command
produces all these subgroups but gives as output only sets of elements
that together with generating elements of U will generate the
subgroups V. Of course, what should have been done is to give as
output a full generating set for each V which can easily be obtained.
Otherwise, the command seems to work correctly and indeed in the case
investigated by Franz G"ahler the only subgroup properly containing
ug0 is the full group. GAP knows in addition to the generating sets
for the subgroups V also their coset tables and that is why it reports
correctly from this knowledge that the only subgroup V, it found, has
index 1.

Unfortunately, the bug is still in GAP 3.4 that has just been
released. It will be corrected in the first patch.

2. Using the command LowIndexSubgroupsFpGroup() on this group with a
presentation on 8 generators and asking for subgroups of index up to
125 is indeed hopeless. Please note that this command works by
searching in a backtrack fashion for homomorphic images of the
finitely presented group in the symmetric group on 125 letters. Even
though that backtrack is rather clever, this is hopeless. Generally,
it should be understood that this command has reasonable chances only
for rather small numbers of generators (2 or 3 say) and that `low
index' is better understood even in most such cases to be well under

3. We do not have special routines for investigating space groups in
GAP as yet. But even though this is not at the top of our priority
list, there are intentions to implement such. Please note that GAP 3.4
provides for the first time the library of all space groups up to
dimension 4 from Brown et al. One way to get certain information is
certainly to follow the suggestion of Franz G"ahler to calculate
modulo certain invariant sublattices of the translation lattice. For
such finite factor groups of the space group one can, for instance,
use faithful permutation representations. In the case of the factor
group of the space group that Franz G"ahler investigates such a
faithful permutation representation can be obtained on the 125 cosets
of the subgroup ug0. A permutation group of degree 125 can, of
course, be very well handled in GAP:

gap> T := CosetTableFpGroup( spg, ug0 );;
#I  CosetTableFpGroup called:
#I      defined deleted alive   maximal
#I      127     2       125     127
gap> Pspg := Group( List( T{[1,3..15]}, t->PermList(t) ), () );;
gap> Size( Pspg );

However, eventually it would indeed be better to create a new kind of
group elements calculating with affine matrices modulo certain
translations as Franz G"ahler suggests.

We would like to mention in this context that at present with some
students we are looking at the task of finding the (Wyckoff classes
of) special positions for a given space group. However, it may take
some time until such functions will be available.

Joachim Neub"user, Werner Nickel.

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