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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

100.

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 ); 7500

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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