October 9th, 2017, 01:03 PM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 101 Thanks: 0  semiproduct
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 $G$. 
October 9th, 2017, 05:17 PM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 89 Thanks: 47 
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>. 
October 9th, 2017, 11:15 PM  #3 
Senior Member Joined: Jan 2015 From: usa Posts: 101 Thanks: 0 
it is $N\rtimes C$, i am sorry

October 10th, 2017, 02:19 PM  #4 
Member Joined: Jan 2016 From: Athens, OH Posts: 89 Thanks: 47 
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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