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October 9th, 2017, 02:03 PM   #1
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semi-product

I need your help to prove the folowing problem:


Let $G$ be a group and $N$ a normal subgroup of $G$ such that $G/N\cong\mathbb{Z}$.


Show that $G=N⋉C$ for some subgroup $C$ of $G$.
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October 9th, 2017, 06:17 PM   #2
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Mona,
This is my last reply to one of your posts until I get some feedback from you on whether my response allowed you to finish the problem. If you can't solve the problem from my hint(s), just ask for more help; otherwise just say "got it".

Let G/N=<xN>. Then it should be easy to prove that G is the semi direct product of N and <x>.
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October 10th, 2017, 12:15 AM   #3
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it is $N\rtimes C$, i am sorry
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October 10th, 2017, 03:19 PM   #4
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Did you not understand that I was saying to set C = <x>, the cyclic subgroup of G generated by x where G/N = <xN>. Here xN is any generator of the cyclic group G/N.
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