[GAP Forum] Zappa-Szep product, knit product

[Burkhard Höfling]

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.