Understanding Semi-Direct Products in Group Theory and Lie Algebras

[haushofer]

Hi,

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 $N_1$ and $N_2$ such that $N_1\cap N_2 = \{e\}$ and $G= N_1 N_2$.

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 :)

 

[haushofer]

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?

 

[Landau]

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

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

You can probably prove the converse for yourself. (just think of {(h,e)|h\in H} and {(e,k)|k\in K})

 

[haushofer]

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! :)