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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??

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1 Answer 1

up vote 3 down vote accepted

$\{0,1\}$ 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=0 –  Natte Dec 29 '12 at 8:49
    
$\{0,a\}$ needs to be stable under addition, that is $a+a \in \{0,a\}$. –  Seirios Dec 29 '12 at 8:52
    
@Natte: In fact, you need the set to be closed under addition so $a+a\in\{0,a\}$. –  B. S. Dec 29 '12 at 9:02
    
So why for n>=3 it's not ok? N=3 z={0,1,2} and 1+1 is fine –  Natte Dec 29 '12 at 9:50
    
@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. –  Seirios Dec 29 '12 at 9:52

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