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April 11th, 2013, 05:01 AM   #1
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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 ? )
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April 12th, 2013, 09:27 AM   #2
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Re: Number of subgroups of infinite abelian group

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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.
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May 27th, 2013, 08:26 AM   #3
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Re: Number of subgroups of infinite abelian group

Thank you Maschke !
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