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Is $\mathbb{Z}_2 \times \mathbb{Z}_2$ is cyclic ?

My attempt :$\mathbb{Z}_2 \times \mathbb{Z}_2$ is not cyclic because gcd$(2,2) \neq 1$

Is it true ?

Shaun
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jasmine
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    Yes I think you are using the lemma that says a finite product of cyclic groups is cyclic if and only if the orders of the factors are all coprime. Alternatively you can inspect the group and see there is no element of order $4$. – Asinomás Aug 11 '21 at 13:29
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    Yes, the lemma @Yorch mentions is overkill. Just check the order of each element. – Kenta S Aug 11 '21 at 13:30
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    Why do you think $\gcd(2, 2)\neq1$ implies it is not cyclic? This is true, but it would be helpful if you explained to us/yourself why it is true, or at least what you think the connection might be. – user1729 Aug 11 '21 at 13:30
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    Why not just write out each element and compute its order? There are only $4$ of them, after all. – lulu Aug 11 '21 at 13:32
  • @user1729 because there are $2$ element in $ \mathbb{Z_2} \times \mathbb{Z_2} $ have order $2$ – jasmine Aug 11 '21 at 13:33
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    @jasmine What does that have to do with being cyclic? (It also isn't true - there are $3$ elements of order $2$). – user1729 Aug 11 '21 at 13:34
  • perhaps that $\mathbb Z_4$ has exactly one element of order $2$, so if you find $2$ it's way too much @user1729 – Asinomás Aug 11 '21 at 13:36
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    @Yorch The point of my comment(s) was to encourage the OP to think and explain themselves, by themselves – user1729 Aug 11 '21 at 13:38
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    @Yorch Ah, it is difficult to be terse, accurate, and unambiguously polite! – user1729 Aug 11 '21 at 13:41

1 Answers1

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For a group of order $n$ to be cyclic, it has to have an element of order $n$. Does this group have such an element?

The main thing that is insufficient with your attempt is that we do not have any idea if you know some relationship between the gcd and whether or not $\mathbb Z_n\times \mathbb Z_m$ is cyclic. It is entirely possible you are grasping at some random fact you do not really know is true or not. You really ought to clearly state what proposition you are leveraging if the proposition is going to be the workhorse of your proof.

rschwieb
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