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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 [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 :)
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 :)