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About semi-direct products

  1. Jan 18, 2010 #1


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    I try to get a grasp on semi-direct products, by notes written by Patrick J. Morandi ("Semi direct products"). I see that the notion of a semi-direct product is more general than a direct product.

    However, the author states that

    A group G is a direct product of two groups iff G contains normal subgroups [itex]N_1[/itex] and [itex]N_2[/itex] such that [itex]N_1\cap N_2 = \{e\}[/itex] and [itex]G= N_1 N_2[/itex].

    Why is this exactly the case?

    And also, how can I translate this for Lie groups on the level of the Lie algebra? (For instance, for the Poincare group). If someone knows good notes or a textbook I'm happy to be informed also :)
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  3. Jan 18, 2010 #2


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    I see that

    N_1\cap N_2 = \{e\}
    gives that the decomposition is unique, but I don't see why the subgroups have to be normal. What happens if they're not?
  4. Jan 18, 2010 #3


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    It is a standard theorem in group theory that if [tex]H[/tex] and [tex]K[/tex] are normal subgroups of [tex]G[/tex] and [tex]H\cap K=\{e\}[/tex], then [tex]HK\cong H\times K[/tex].

    see e.g. http://homepage.mac.com/ehgoins/ma553/lecture_21.pdf [Broken] ("recognition theorem").

    You can probably prove the converse for yourself. (just think of {(h,e)|h\in H} and {(e,k)|k\in K})
    Last edited by a moderator: May 4, 2017
  5. Jan 18, 2010 #4


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    Ok, thanks! Yes, the converse is quite clear to me I guess, but I don't see clearly why these subgroups have to be normal. I'll check your link, thanks again! :)
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