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I know there are infinitely many proper subgroups of $\mathbb{Z} \oplus \mathbb{Z}$, but is $\mathbb{Z} \oplus \mathbb{Z}$ cyclic?

Arturo Magidin
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Harry
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  • So... you have a title to your question, but the first thing you say is that you already know the answer. Why put a title that is not what you are actually asking, but something else entirely? Are you purposely trying to mislead people? – Arturo Magidin Oct 05 '19 at 06:06
  • No I never want to mislead anyone. forgive me... I am new Contributor. Please edit if anything sees wrong. – Harry Oct 05 '19 at 06:10
  • It shouldn’t take too much experience to realize that asking a question in the title that is not what your actual question is will definitely be misleading. – Arturo Magidin Oct 05 '19 at 06:22
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    Hint: Every nontrivial sugroup of a cyclic group has finite index. – Arturo Magidin Oct 05 '19 at 06:23

1 Answers1

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Take a and b different integers.

Easy way to think is

If there is any generator, it can be either of the form 1)(a,a) or 2)(a,b)

If it is of form 1, you can never obtain element of second form.

Can you finish the argument?

ogirkar
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