[GAP Forum] Zappa-Szep product, knit product

Fri Dec 29 23:13:36 GMT 2006

Dear Dr. Pride,

thank you for your example. There are certainly many more situations giving 
rise to Z - S products. It is clear, for example, that all semidirect 
products and even the direct products are special cases of Z-S products. 
Among others in this ongoing discussion it is to show, that Z-S products 
really make sense in concrete cases, to evaluate for a possible even partial 
implementation in GAP.  So your example, I will evaluate later on after 
reading the referenced texts, is welcome indeed.

thank you for your answer, kind regards, Rudolf Zlabinger

----- Original Message ----- 
From: <sjp at maths.gla.ac.uk>
To: "GAP Forum" <forum at gap-system.org>; "Burkhard Höfling" 
<burkhard at hoefling.name>; "Rudolf Zlabinger" <Rudolf.Zlabinger at chello.at>
Sent: Friday, December 29, 2006 9:47 PM
Subject: Fwd: Re: [GAP Forum] Zappa-Szep product, knit product

> 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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