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
<[EMAIL PROTECTED]> -----
Date: Fri, 29 Dec 2006 18:13:27 +0100
From: Rudolf Zlabinger <[EMAIL PROTECTED]>
Reply-To: Rudolf Zlabinger <[EMAIL PROTECTED]>
Subject: Re: [GAP Forum] Zappa-Szep product, knit product
To: GAP Forum <[EMAIL PROTECTED]>, Burkhard Höfling
<[EMAIL PROTECTED]>
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" <[EMAIL PROTECTED]>
To: "Rudolf Zlabinger" <[EMAIL PROTECTED]>
Cc: "GAP Forum" <[EMAIL PROTECTED]>
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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