Fwd: Re: [GAP Forum] Zappa-Szep product, knit product

[sjp at maths.gla.ac.uk]

Hi, The following situation has been looked at, and is quite reasonable to work
with:

If A is generated by X, and B generated by Y, then the action of X sends Y to
Y, and the action of Y on X sends X to X^*. It is then fairly easy to write
down the conditions for the actions do give rise to a Z-S product. See:

M. G Brin, On the Zappa-Szep product, Commun in Algebra, 33 (2005), 393-424

See also:
T. G. Lavers, Presentation of general products of monoids, J Algebra, 204
(1998), 733-741

                             Steve Pride



----- Forwarded message from Rudolf Zlabinger <Rudolf.Zlabinger at chello.at> -----
    Date: Fri, 29 Dec 2006 18:13:27 +0100
    From: Rudolf Zlabinger <Rudolf.Zlabinger at chello.at>
Reply-To: Rudolf Zlabinger <Rudolf.Zlabinger at chello.at>
 Subject: Re: [GAP Forum] Zappa-Szep product, knit product
      To: GAP Forum <forum at gap-system.org>, Burkhard Höfling
<burkhard at hoefling.name>

Dear Dr. Höfling,

thank you for your detailed answer.

Originally I had no specific application in mind, my question was solely
about the existence of code implementing the Zappa Szep product in general.
In details:
1. Code that supports the finding of functions g and h, that satisfy the
required properties of the definition. 2. Code to produce a representation
of the product itself.

So presently I have no concrete groups and functions in mind. I agree, that
such methods would only support relatively small groups, comparable that for
the semidirect products, if one awaits a satisfacting efficiency.

I agree also, that it should be possible to define the functions in terms of
the groups generators, as there are rules for the multiplication in the
functions definitions, paid by loss of efficiency.

I also agree, that it may not feasable to determine the functions fulfilling
of the conditions of the definition by computational algorithms.

There, indeed, is no theoretical problem to determine the internal factors
of a Zappa -Szep product, the challenge is the external form of the product.
Nevertheless thank you for your suggestions.

To sum up your message:

It may be not feasable to implement the first part of the Zappa-Szep product
by computational methods: the finding of suitable functions g and h. The
second part, the production of a presentation of the product should be
possible, if the functions g and h are given. If the functions are given in
terms of generators, it may be done with loss of efficiency.

So I conclude for my original question: There is, presently, no code known
to the forum supporting the whole or parts of the Zappa-Szep product. There
are reasons for, as computablity of parts of this construct seems not
sufficiently to be given in general.

thank you again for answering me, kind regards, Rudolf Zlabinger


----- Original Message -----
From: "Burkhard Höfling" <burkhard at hoefling.name>
To: "Rudolf Zlabinger" <Rudolf.Zlabinger at chello.at>
Cc: "GAP Forum" <forum at gap-system.org>
Sent: Friday, December 29, 2006 3:54 PM
Subject: Re: [GAP Forum] Zappa-Szep product, knit product


> Dear Dr Zlabinger,
>
>> I found a short description of the Zappa Szep product in the  following
>> link:
>>
>> http://en.wikipedia.org/wiki/Zappa-Szep_product
>>
>> In the link there are also references to related textbooks.
>
> thanks for sending the above explanation. However, I am still unsure  what
> applications of the Zappa Szep product you have in mind.
>
> - Do you have concrete groups H and K, and explicit (GAP) functions h  and
> k having the properties given in the definition of an external  Zappa Szep
> product? This would be fairly easy to implement, but would  only work
> reasonably efficiently for relatively small groups (the  same problems
> arise for seimidirect products as well). If this is  what you are
> interested in, what are the orders of H and K that you  have in mind?
>
> - In principle, it would be sufficient to define functions h and k in
> terms of generators of H and K only. This would be possible as well,  but
> efficiency would be generally worse than in the first case. In  fact, you
> could use this to write down a presentation (even a  rewriting system) for
> the product, given presentations (rewriting  systems) of H and K.
>
> Note that in both cases, it would be nearly impossible to tell if h  and k
> indeed satisfy the properties required by the definition of the  Zappa
> Szep product.
> In particular, I don't think that it would be computationally  feasible to
> list all possible Zappa Szep product of two given groups,  except for
> ridiculously small examples.
>
> - Or you may actually be interested if a given group is the Zappa  Szep
> product of two subgroups. In this case, one cannot, in my  opinion, do
> much better than to compute the subgroup lattice and look  at pairs of
> subgroups such that the product of their orders is the  group order and
> which intersect trivially. Note that it is enough to  look at conjugacy
> class representatives of subgroups - if G is the  Zappa Szep product of H
> and K, then it is also the product of H^g1  and K^g2 for all g1, g2 in G.
>
>
>
>
>


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