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 April 11th, 2013, 05:01 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, 09:27 AM   #2
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Re: Number of subgroups of infinite abelian group

Quote:
 Originally Posted by thippli 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 ? )
Yes. If the group contains a copy of the integers (meaning it has a subgroup isomorphic to the integers) then that subgroup contains infinitely many subgroups, just as the integers do.

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, 08:26 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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