Semi-Direct Product: Explained in Plain English

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In summary: NilleHecke algebras?This is a discussion about the semi-direct product, which is an operation between two groups. The most important group that someone knows of is the Poincaré group, which is the semidirect product of the abelian translation group generated by P^\mu and the rotation group generated by J^{\mu\nu}. Composing the two groups in the obvious way gives the (b,n) equation. Other examples of semi-direct products are the translations and scales of the real line, and the actions of the real line.
  • #1
nille40
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Hi!
Could someone please explain to me what semi-direct product is, in plain english? English isn't my native language, so I would really appreciate if mathematical lingo could be avoided.

I would also appreciate some online resources. I've tried www.mathworld.com,[/URL] but that wasn't very easy to understand...

Thanks in advance,
Nille
 
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  • #2
I can only think of the dot-product for vectors. If you have two vectors:
[tex]A= \left(\begin{array}{c}A_x\\ A_y\\ A_z\end{array}\right)[/tex]
and
[tex]B= \left(\begin{array}{c}B_x\\ B_y\\ B_z\end{array}\right)[/tex]

Then the dot-product will be:
[tex]A\cdot B= \left(\begin{array}{c}A_x B_x\\ A_y B_y\\ A_z B_z\end{array}\right)[/tex]

Hope this helps!
 
  • #3
Thanks for responding!
However, it is not the dot-product I am looking for.

A semi-direct product is an operation between two groups. And that's pretty much all I know :)

Nille
 
  • #4
Are you familiar with normal and quotient groups?
 
  • #5
http://dominia.org/djao/semidirect.pdf
 
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  • #6
Thanks! Actually, I think I've read that document, and thus failed to understand it... :)

Do you have any suggestion on where I should start reading? I've read about homo- and automorphisms, and I understand these pretty well. It's the final step, on which I need a clear definition, preferably in natural language, as opposed to mathematical language.

As for normal and quoutien groups - I have no idea. I'm learning (unfortunatly) the swedish lingo, so it's hard to say.

Thanks!
Nille
 
  • #7
perhaps it would be useful to look at an example.

the most important one that i know of is the Poincaré group, which is the semidirect product of the abelian translation group generated by [itex]P^\mu[/itex] and the rotation group generated by [itex]J^{\mu\nu}[/itex]. composition is defined in the obvious way.

or an even simpler example, consider the following actions of the real line.

translations:
[tex]
\begin{align*}T(a):&\mathbb{R}\longrightarrow\mathbb{R}\\
&x\longmapsto x+a
\end{align*}
[/tex]

and scales:
[tex]
\begin{align*}
S(m):&\mathbb{R}\longrightarrow\mathbb{R},\quad m\neq 0\\
&x\longmapsto mx
\end{align*}
[/tex]

the first one makes a group that is isomorphic to the reals under addition, and the second one is a group that is isomorphic to the reals without zero, under multiplication.

you can compose the two in the obvious way:

[tex]
(b,n)(a,m)=(na+b,nm)
[/tex]
which we find from the equation [itex]n(mx+a)+b=nmx+na+b[/itex]

notice that if instead we had an honest to goodness direct product on our hands, the rule would be
[tex](b,n)(a,m)=(a+b,nm)[/tex]
 
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  • #8
Thanks! That's probably the best description of semi-direct product I've read! And I've scanned the larger part of the internet...:)

Thank you very much!
Nille
 
  • #9
Originally posted by nille40
Thanks! That's probably the best description of semi-direct product I've read! And I've scanned the larger part of the internet...:)

Thank you very much!
Nille
i hope that helped. the point here is that instead of two copies of a group that act independently as they would in a direct product, you get ordered pairs where one of the groups acts normally, and the other group acts through some homomorphism on the first group.

the example i gave above is very natural and easy to understand, so read the paper linked above with this example in mind, and try to associate all the objects described in that paper with the translations and scales i described above.
 
  • #10
Originally posted by lethe
perhaps it would be useful to look at an example.

the most important one that i know of is the Poincaré group, which is the semidirect product of the abelian translation group generated by [itex]P^\mu[/itex] and the rotation group generated by [itex]J^{\mu\nu}[/itex].

if that's the most obvious/important one you know you ought to get out more! the simplest one is of course the semi direct product of C_2 and C_3 that is not C_6 ie it is S_3

note tongue firmly planted in cheek, but in general A_n semi direct prod with C_2 is fairly important!
 
  • #11
Originally posted by matt grime
if that's the most obvious/important one you know you ought to get out more! the simplest one is of course the semi direct product of C_2 and C_3 that is not C_6 ie it is S_3

note tongue firmly planted in cheek, but in general A_n semi direct prod with C_2 is fairly important!

well, i think the Poincaré group is the most important group in all of physics, and is therefore more important than S6, but i suppose it is a matter of opinion, so choose whatever group you want.
 
  • #12
Originally posted by lethe
well, i think the Poincaré group is the most important group in all of physics, and is therefore more important than S6, but i suppose it is a matter of opinion, so choose whatever group you want.

What about the Lorentz group? Or the thingy bob group associated to a manifold (pontrjagin? picard?) How's the affine transformations most important? Serious question, unlike my previous answer, which was genuinely facetious. I mean what does it control? This was posted in linear algebra not physics, and here the Poincare group might be less important than all the permutation groups (which are finite), and therefore Hecke algebras. And it's a small step from there via duality theroems to objects of great importance in theoretical physics, but that's a whole new topic (in Lie theory etc)
In fact we get all D_n as semidirect products too. Indeed as the Poincare group deals in affine transformations and not linear ones... ok tongue still in cheek for that one. Any takers to explain PSL_3(F_2)?
 
  • #13
Originally posted by matt grime
What about the Lorentz group? Or the thingy bob group associated to a manifold (pontrjagin? picard?) How's the affine transformations most important?
yes, like i said, it is a matter of opinion, so take whichever you want and call it the most important.
Serious question, unlike my previous answer, which was genuinely facetious. I mean what does it control? This was posted in linear algebra not physics, and here the Poincare group might be less important than all the permutation groups (which are finite), and therefore Hecke algebras.
it is a good point. i might argue that the Poincaré group is the most important group in physics, but it is probably not such an important group to a mathematician.


And it's a small step from there via duality theroems to objects of great importance in theoretical physics, but that's a whole new topic (in Lie theory etc)
In fact we get all D_n as semidirect products too. Indeed as the Poincare group deals in affine transformations and not linear ones... ok tongue still in cheek for that one. Any takers to explain PSL_3(F_2)?
what is F2?
 
  • #14
F_2 is (here) the field with two elements, PSL_3(F_2) is the symmetries of the fano plane which I seem to have some memory as being Octonionic and therefore interesting in the John Baez view point on Mathematical physics.
 
  • #15
Originally posted by matt grime
F_2 is (here) the field with two elements
thanks. i am too used to seeing Z2, but i guess i should have known from context.
 

FAQ: Semi-Direct Product: Explained in Plain English

What is a semi-direct product?

A semi-direct product is a mathematical concept that describes a group formed by combining two smaller groups in a specific way. It is a type of group extension, where one group is "semi-directly" embedded into the other.

How is a semi-direct product different from a direct product?

In a direct product, the two smaller groups are completely independent and do not interact with each other. In a semi-direct product, there is some interaction between the two groups, but it is not as strong as in a direct product.

What is the significance of a semi-direct product in mathematics?

Semi-direct products are important in studying group theory, a branch of mathematics that deals with the algebraic structures of groups. They provide a way to understand and classify different types of groups.

Can you explain the concept of a semi-direct product in plain English?

Imagine two groups, A and B, as two puzzle pieces. In a direct product, these pieces are simply placed side by side, with no overlap or interaction. In a semi-direct product, one piece is placed on top of the other in a way that they partially overlap, but they are not completely glued together.

How is a semi-direct product calculated?

A semi-direct product can be calculated using a specific formula, called the semi-direct product formula. This formula involves finding a homomorphism (a type of function) between the two groups, which determines how they interact and combine in the semi-direct product.

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