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

[Hulpke,Alexander]

Dear Martin Rubey, Dear Forum,

Let me briefly comment on some aspects of your plan: 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)?

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

- In the end very few groups are products in a meaningful way. You might have 
  a) multiple product decompositions that look different but yield the same group
  b) product decompositions that look very similar (e.g. iterated semi direct products) but yield a large number of different groups.
  c) Many groups that do not have a product decomposition, but only a description as an extension.

If you look at groups up to order 20 or so, it seems that there are multiple series that have plausible names (quasi-semi-dihedral and so on) , but once you go to order 50-100 all of this explodes — you simply cannot define good names arbitrarily far.

This basically means that you cannot expect to have a general routine to find a “construction name"

Instead for a description, probably the best case is to use as name an identification in the small groups or transitive groups or primitive groups libraries if the groups are in, and some ad hoc names for e.g. wreath products in various actions.

You could restrict to groups that you can construct in a nice way, but then you will get asymptotically 0% of all groups.

Concerning a recognition of $C_m\wr S_n$, test:
- The radical is elementary abelian of dimension $n$.
- 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.

Regards, Alexander Hulpke

> On Dec 16, 2017, at 2:00 PM, Rubey Martin <martin.rubey at tuwien.ac.at> wrote:
> 
> Dear all,
> 
> I was directed to this forum by Derek Holt, essentially I am migrating two mathoverflow questions.  I must admit that I have practically no knowledge of group theory.  
> 
> BACKGROUND (probably not necessary to answer the questions):
> 
> The reason for my interest is that I would like to make "(isomorphism classes of) finite groups of small order" into a new collection for the database http://findstat.org.  This would enable findstat's search engine, similar in spirit to the http://oeis.org, to recover a given group parameter, eg. the number of conjugacy classes, given the values of the parameter on a few small groups.  More interestingly, it frequently turns out that a strange parameter on one kind of objects (eg., finite graphs), is rather easy to understand on other objects obtained by applying a natural map (eg., the automorphism group of a graph).  Findstat would then discover this, too.
> 
> The main difficulty in establishing finite groups as a collection for findstat is the lack of a canonical representation for finite groups: to make the search engine efficient, it is necessary that every object in the collection is uniquely represented by a (short) string.  This canonical representation has to allow the reconstruction of the group - I am just realising that this could actually be circumvented, although not easily.  Ideally, the canonical representation would be human readable, within limits.
> 
> Although desirable, it is not strictly necessary that every object has a canonical representation, it is fine if for some groups the algorithm aborts.
> 
> QUESTIONS (the first question is a bit vague, the second question is rather specific and mostly independent of the first):
> 
> A. Although I have a mostly working - albeit naive - design to obtain a canonical representation for sufficiently many groups meanwhile, I am very interested in your comments.  What I do is the following:
> 
> 1. decompose the group (given as a permutation group) into a direct product (using DirectFactorsOfGroup),
> 2. for each factor, check a few special cases, in this order: IsCyclic, IsAlternatingGroup, IsSymmetricGroup, IsDihedralGroup, IsQuaternionGroup, IsQuasiDihedralGroup, IsPSL, IsSL, IsGl, and finally, all else failing, whether the group has an IdSmallGroup.
> 
> Does this sound reasonable, or is there a much better scheme?  Am I leaving out any obvious constructions?
> 
> In any case, doing so, all automorphism groups of graphs on at most 8 vertices are covered.  By contrast, the groups associated with finite Cartan types (which is another collection in findstat), are only covered for very small rank.  This leads to my second question:
> 
> B. I thought it would be useful to check also the special case that the group is a "generalized symmetric group", that is, the wreath product of a cyclic and a symmetric group.  Derek Holt suggested to do the following:
> 
> To check that G is of the form C_m § S_n:
> 1. compute the  largest solvable normal subgroup F of G (using RadicalGroup), check that F is Abelian
> 2. compute the quotient Q = G/F, check that its order is m^n n!
> 3. compute the Abelian invariants of the Fitting subgroup to check that it is C_n^m
> 4. check that quotient Q is S_n
> 
> Now, there are two problems that remain:
> 
> a) what should I do for n=3 and n=4?  One thing that might work is to compute m (how?), and then check for isomorphism.  Is there a better way?
> 
> b) given that I have almost no knowledge of group theory, why does the above work?
> 
> It is plausible to me that it recognises C_m^n § S_n for n > 4, because in these cases S_n is not solvable and C_m^n is a normal solvable subgroup. However, it is not clear to me why C_m^n is the larges normal solvable subgroup.
> 
> It is even less clear to me, why there are no other groups with quotient S_n and largest solvable normal subgroup C_m^n...
> 
> Many thanks for your help!
> 
> Martin Rubey
> TU Wien
> 
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