[GAP Forum] Canonical form for some small groups and efficient characterisation of the generalized symmetric groups

[Thomas Breuer]

Dear Martin,

coming back to an initial question asked by Alexander,
your examples seem to indicate that *isomorphism as abstract groups*
is not the appropriate notion of equivalence.

When groups arise as symmetries of finite sets such as the vertices
of graphs then it is more natural to consider *permutation isomorphism*
(that is, conjugacy in the symmetric group on the given points).
For example, a group of order two can act on four points
by swapping two pairs or by fixing two points and swapping the other
two points; these two possibilities should probably be distinguished
in such a context.

With respect to permutation isomorphism, groups are considered as small
when they are permutation groups on a small set, regardless of their
group orders.
GAP's library of transitive groups provides a reasonable source of
small groups in this sense.

All the best,
Thomas


On Sun, Dec 17, 2017 at 08:58:45AM +0100, Martin Rubey wrote:
> Dear Alexander Hulpke, Dear Forum,
> 
> many many thanks for your comments!  Let me try to clarify - I apologize
> for the lengthy text...
> 
> > There is no fundamental obstacle, but you either will end up with just
> > referring to some of the libraries of groups, or end up with an
> > exceeding amount of work by hand to make things come out nicely:
> >
> > - What groups are you planning to classify? Abstract groups or
> >   Permutation groups (i.e. group actions)?
> 
> the idea is to have finite abstract groups in findstat.
> [...]