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I would like to ask if $G$ is a semidirect product of a normal subgroup $N$ by a subgroup $K$.

Can we consider $N\wedge G$ as a subgroup of $G\wedge G$? In fact, is the following sequence exact $0\rightarrow N\wedge G\rightarrow G\wedge G \rightarrow G/N\wedge G/N \rightarrow 0$?

yours,

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You will have to explain your question and cryptic notation for anyone to make progress on this. –  rschwieb Aug 22 '12 at 16:05
    
Have you looked at this yet? –  Alexander Gruber Aug 23 '12 at 5:44
    
Have you also tried using the fact that semidirect products are defined by extensions which split? –  Ronnie Brown Aug 23 '12 at 9:48
    
For a moment I thought the symbol $\,\wedge\,$ within this context could mean intersection, but $\,G\wedge G\,$ would be boringly trivial, so...?? –  DonAntonio Aug 25 '12 at 3:52
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