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April 11th, 2013, 06:01 AM  #1 
Newbie Joined: Oct 2012 Posts: 4 Thanks: 0  Number of subgroups of infinite abelian group
Is the number of subgroups of an infinite abelian group always infinite ? ( or Is there any infinite abelian group having only finite number of subgroups ? ) 
April 12th, 2013, 10:27 AM  #2  
Senior Member Joined: Aug 2012 Posts: 2,156 Thanks: 630  Re: Number of subgroups of infinite abelian group Quote:
If not, then every element of the group has finite order; in which case each element generates a cyclic subgroup of finite size. Since the group is infinite, there must be infinitely many cyclic subgroups.  
May 27th, 2013, 09:26 AM  #3 
Newbie Joined: Oct 2012 Posts: 4 Thanks: 0  Re: Number of subgroups of infinite abelian group
Thank you Maschke !


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