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

[Martin Rubey]

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.

> - Is your goal to classify *all* groups up to some parameter, or just
>   some?

I am guessing that it is essentially impossible to classify all groups.
Fortunately, for the purposes of findstat this is not nessecary.
Asymptotically 0% of all groups is not a problem.

Indeed, what I would have liked most is to simply use the small groups
library IdSmallGroup.  Unfortunately, this does not quite work, let me
explain:

For each collection of objects in findstat, we consider some objects
"small".  For example simple graphs of at most 6 vertices, finite posets
on at most 5 elements, permutations of 1,..,5, integer partitions of at
most 10, finite Cartan types of rank at most 8, etc.  A finite group of
order at most 47 will be "small".  I need to be able to uniquely
identify groups these "small" groups.

The findstat database also contains maps between collections.  For
example, there will be a map "automorphism group" from graphs to groups,
a map "Weyl group" from finite Cartan types, a map "center" from groups
to groups, etc. [besides: I could not come up with all that many
interesting maps from groups to other objects, like permutations, yet]

Thus, I additionally need to uniquely identify those finite groups that
occur after applying a few (2 or 3) maps as image of some other small
object.  For example, applying "automorphism group" to the complete
graph on 6 vertices, or "Weyl group" to A_5 we get the symmetric group
S_6, so this group needs to be uniquely identified.

Being able to identify these images is not an absolute requirement - if
some group occurs as an image and the "classification algorithm" fails,
that's OK.  However, it has to be an isomorphism invariant, and it
shouldn't fail "too often".  Also note that maps may be added in the
future, so some level of robustness is necessary.

> Concerning a recognition of $C_m\wr S_n$, test:

> - The radical is elementary abelian of dimension $n$.

except for n=3 and n=4, right?

> - The radical has (you need special treatment for 6) a complement
> that is a split extension of $A_n$ with $C_2$, You can recognize
> $A_n$ as being a simple group of order $n!$.

> - If $U$ is a point stabilizer of $S_n$, check that an eigenvector of
> $U$ for eigenvalue 1 has an orbit of length $n$ and this orbit forms
> a base — thus it is the natural permutation module.

I think that's what GAP's IsSymmetricGroup does, right?


Many thanks again,

Martin (Rubey)