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I need to find all the values of $a$ and $n$ that gives $\{0,a\}$ is a subgroup of the group $(\mathbb{Z}_n,+)$. Assum $n \geq 2$ and $a \neq 0$.

Actually I thing that for each $a$ and $n$ we will get that this is a subgroup ($0$ identity element) Am I wrong??

1 Answers 1

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$\{0,1\}$ is not a subgroup of $\mathbb{Z}_n$ for $n \geq 3$.

Hint: Notice that $\{0,a\}$ is a subgroup of $\mathbb{Z}_n$ iff $2a=0$.

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    I don't understand why 2a=02012-12-29
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    $\{0,a\}$ needs to be stable under addition, that is $a+a \in \{0,a\}$.2012-12-29
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    @Natte: In fact, you need the set to be closed under addition so $a+a\in\{0,a\}$.2012-12-29
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    So why for n>=3 it's not ok? N=3 z={0,1,2} and 1+1 is fine2012-12-29
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    @Natte: $\{0,1\}$ is not a subgroup of $\mathbb{Z}_3$ because $2=1+1 \notin \{0,1\}$: $\{0,1\}$ is not stable under addition.2012-12-29
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    Ok thanks I was little confused so there is no a that will make this set a subgroup2012-12-29
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    @Natte: In fact, there is always $a=0$. Otherwise, you have to investigate when $\mathbb{Z}_n$ has an element of order $2$.2012-12-29
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    I was given that a is not zero so...Thank you very much for your help2012-12-29