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

[Colva Roney-Dougal]

Dear Martin

Can I suggest as a canonical form for a conjugacy class of permutation group G \leq S_n

1. Fix once and for all an ordering on the permutations in S_n - lex ordering when storing the permutations as their image lists would do.
2. Find the minimal number of generators d of G.
3. Store a lex-minimal generating set amongst the different ones of size d?

I’m not completely sure this would work, and it would be awkward to compute, but it would give a canonical representative of each conjugacy class of groups?

Best wishes

Colva

> On 18 Dec 2017, at 12:10, Thomas Breuer <sam at Math.RWTH-Aachen.De> wrote:
> 
> 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.
>> [...]
> 
> 
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*************************************

Colva Roney-Dougal
Reader in Pure Mathematics
Director of the Centre for Interdisciplinary Research in Computational Algebra
Editor of Proceedings A of the Royal Society of Edinburgh
Editor of Royal Society Open Science
Director of Undergraduate Mathematics Admissions

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