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If I am asked to "Construct explicitly a non-commutative semi direct product..", does that mean that because they have semd "non commutative", I don't include the trivial SDP, i.e the direct product? Or I include this and any other SDPs I get.

EDIT: Also, probably a silly question, but there ALWAYS exists the direct product as an SDP doesnt there?

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There does not always exist a nontrivial Semidirect product. –  Mr.Guy Jan 11 '13 at 23:47
@Mr.Guy Oh ok, so I basically have to state the direct product and the nontrivial one, assuming it exists. If it doesn't then that makes my life easier –  Kaish Jan 11 '13 at 23:49
It sounds as if you only need to construct one thing, a semi direct product that is not commutative (which means you would not do the direct product), or am i understanding the question wrong? –  Mr.Guy Jan 11 '13 at 23:50
@Mr.Guy This is the confusion I'm having. The question literally say "Construct explcitley a non-commutative semi direct product $H \rtimes Q$ with.." and then it gives me what the groups are. –  Kaish Jan 11 '13 at 23:51
I think the question is explicitly including "noncommutative" because the point is not to give the (trivial) semidirect product that is just the direct product. So I'd say you should not bother including the direct product. If you were asked to give a noncommutative group of order 6, writing down ${\mathbf Z}/(6)$ is not at all relevant to an answer. –  KCd Jan 11 '13 at 23:52

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