Re: Zappa-Szep product in wiki

In article <1185017811.231477.252610@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Anvita <anvita21@xxxxxxx> wrote:
On 20 jul, 15:14, magi...@xxxxxxxxxxxxxxxxx (Arturo Magidin) wrote:
In article <1184954680.014513.202...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,

Anvita <anvit...@xxxxxxx> wrote:
The definition of the Zappa-Szep product that is given in wikipedia at

http://en.wikipedia.org/wiki/Zappa-Szep_product

includes the property

[...]

b(k, h1*h2) = b(b(k,h2),h1).

However, I suspect that it should instead be

i.e., b(k, h1*h2) = b(b(k,h1),h2).

because, for example, if K normal in G,
we have k h =3D h (k^h)

k*h = h*(k^h)

(this holds regardless of whether K is normal...)

It surely does. But if K is not normal then k^h is in general not in
K which is not what we want.

"We", Kimosabe?


and so =CE=B2(k, h)=3Dk^h.

k^(h1*h2) = (k^h1)^h2

Again: this holds regardless of whether K is normal: for all elements
a, b, and c in a group

a^(bc) = (bc)^{-1} a (bc) = c^{-1}b^{-1}abc = c^{-1}(b^{-1}ab)c
= c^{-1}(a^b)c
= (a^b)^c.

Hence,
the left-hand side of the above relation is

k^(h1 h2) - (k^h1)^ h2
which equals
b(b(k,h1),h2).
Am I right?

Not entirely. Instead of using the operation x^y = y^{-1}xy, you can
use the operation

beta(k,h) = hkh^{-1} = {}^h k.


This is still not clear to me.

You asked about defining b (beta) via conjugation. There are TWO
conventions for conjugation-by-h. Either x |-> hxh^{-1}, or
x|-> h^{-1}xh.

In a sense, the two are equivalent: it's just a question of left-right
symmetry. You want to use the second convention, which leads to an
equation involving b which is not the one in Wikipedia. But if you
stick to the FIRST convention, then the equation in Wikipedia is the
correct one that codifies that convention, as opposed to the version
you want.


If we define b ( =beta ) this way then,
in order for the relation

k*h=a(k,h)*b(k,h)

To have this equation, you are assuming that G=HK is an internal
Zappa-Szep product, which requires in addition that H/\K = {1}. So
what we have is a semi-direct product of K by H. In that situation,
you want to define the maps the ->other<- way, i.e., with the normal
group playing the role of H in the definitions in the Wikipedia page,
and the not-necessarily normal playing the role of K.


to hold, we should define a(k,h)=k*h^2*k^{-1}*h^{-1}.

You do realize that you are transposing the roles of H and K in an
internal product? When G is a semidirect product of two subgroups, the
realization of the Zappa-Szep product assumes that H is normal and K
acts on H, not the other way around; the realization in the wikipedia
article (under "Relation to semidirect and direct products") takes
a(k,h)=khk^{-1} (notice the conjugation convention), and
beta(k,h)=k.

What exactly are you trying to do?

Try starting over and switching your conjugations from x|-> h^{-1}xh
to x|->hxh^{-1}. The equations should come out correctly then.



--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

Relevant Pages

  • Re: Zappa-Szep product in wiki
    ... Anvita wrote: ... equation involving b which is not the one in Wikipedia. ... stick to the FIRST convention, then the equation in Wikipedia is the ... You are not saying that in a Zappa-Szep product G=HK the subgroup K ...
    (sci.math)
  • Re: Zappa-Szep product in wiki
    ... equation involving b which is not the one in Wikipedia. ... stick to the FIRST convention, then the equation in Wikipedia is the ... You are not saying that in a Zappa-Szep product G=HK the subgroup K ... which factor is normal or which "conjugation convention" one takes. ...
    (sci.math)
  • Re: Member Variables Naming Convention
    ... Regardless of the naming convention used in a ... If the convention ... preference but no tales of confusion have been told. ...
    (microsoft.public.dotnet.languages.csharp)
  • Re: Member Variables Naming Convention
    ... Regardless of the naming convention used in a ... If the convention ... It wasn't the OP but a followup that mentioned ... preference but no tales of confusion have been told. ...
    (microsoft.public.dotnet.languages.csharp)
  • Re: Member Variables Naming Convention
    ... Regardless of the naming convention used in a ... If the convention ... preference but no tales of confusion have been told. ...
    (microsoft.public.dotnet.languages.csharp)