[GAP Forum] Retrieving group name

[Bettina Eick]

Dear Gap-forum,

Igor Schein wrote:
> I am struggling to find the way to retrieve canonical group names in
> GAP4 ( I've done it once in GAP3 ).  Basically, I need a function
> which will output "Q8" if the input is SmallGroup(8,4), for example.

In GAP 3 the catalogue of all solvable groups of order at most 100 contained
names for many of the groups in this catalogue. I guess that Igor refers to
these names.

In GAP 4 this catalogue of solvable groups is not explicitly available
anymore as all the groups in the catalogue are contained in the small
groups library. The relation between the two catalogues is given by the
GAP 4 function Gap3CatalogueIdGroup. But the names of the groups as in
GAP 3 have not been translated to GAP 4. The basic reason for this is
that Hans Ulrich Besche (who implemented the GAP 3 names) considered the
GAP 3 names as too unsystematic.

The GAP 3 names are based on the well-known names for certain groups such
as the cyclic, dihedral, symmetric and alternating groups. Beyond these
groups, the GAP 3 names use a structure analysis to determine a name; for
example, An.C2 would be a group with An as normal subgroup of index 2. Thus
the GAP 3 names are just a description of the group structure.

The GAP 3 names should not be considered as 'canonical'. Also, there are
groups of order less than 100 which did not get a name under this scheme,
as there did not exist a nice, short and unique structure analysis that
could be used as a name. Further, some groups could have two or more
different names, as more than one structure analysis applies to them.

Nonetheless, a more systematic version of the GAP 3 names would perhaps be
nice to have in GAP 4 also. I'll think about it.

Determining canonical names for all finite groups is a different project.
'Canonical' means to me that every isomorphism type of group gets a unique
name such that non-isomorphic groups have different names. Thus a canonical
name really is a description of the group up to isomorphism. For such a
naming scheme we need at least as many names as we have isomorphism types
of groups. As the number of groups of order n grows exponentially with the
largest exponent in the prime-power factorisation of n, it is obvious that
we cannot expect canonical names to be short and easy.

For p-groups there exists the standard presentation algorithm in the ANUPQ
package by Eamonn O'Brien. This can determine a standard presentation for
a p-group G. Such a presentation is unique for the isomorphism type of G and
thus it could be considered as a canonical name for G. If G has order p^n,
then a standard presentation has n(n+1)/2 relations and each relation can
be written as a vector of length at most n over GF(p). Thus a standard
presentation is quite long and perhaps not easy to read as name for the
group. But considering the number of groups of order p^n, it seems unlikely
that one can get much shorter canonical names in general.

I think this idea of computing a standard presentation extends at least to
all finite solvable groups. This extension has not been implemented so far,
but it would be a very worth-while project. If we have such a function that
can determine a canonical name or a standard presentation for a finite group,
then this would also solve the isomorphism problem and thus would be very
valuable. If anybody is interested in following this project up, then please
let me know. I would be happy to supply further information.

Best wishes,
            Bettina