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Semidirect product 
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#1
Dec103, 01:59 PM

P: 34

Hi!
Could someone please explain to me what semidirect 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, but that wasn't very easy to understand... Thanks in advance, Nille 


#2
Dec203, 04:51 AM

P: 265

I can only think of the dotproduct 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 dotproduct 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
Dec203, 05:02 AM

P: 34

Thanks for responding!
However, it is not the dotproduct I am looking for. A semidirect product is an operation between two groups. And that's pretty much all I know :) Nille 


#4
Dec203, 06:57 AM

P: 333

Semidirect product
Are you familiar with normal and quotient groups?



#5
Dec203, 07:32 AM

P: 265



#6
Dec203, 09:12 AM

P: 34

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
Dec203, 11:50 AM

P: 657

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] 


#8
Dec303, 03:57 AM

P: 34

Thanks!! That's probably the best description of semidirect product I've read! And I've scanned the larger part of the internet...:)
Thank you very much! Nille 


#9
Dec303, 04:06 AM

P: 657

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
Jan1604, 05:23 PM

Sci Advisor
HW Helper
P: 9,396

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


#11
Jan1604, 05:51 PM

P: 657




#12
Jan1604, 06:30 PM

Sci Advisor
HW Helper
P: 9,396

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
Jan1604, 06:56 PM

P: 657




#14
Jan1704, 05:19 AM

Sci Advisor
HW Helper
P: 9,396

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.



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