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I know that $ (\mathbb{Z}, +) $ is a cyclic group. It is generated by $1$ or by $-1$.

But who generates the $0$ (zero) element?

Thank you!!!!

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    So when we are generating the set, we have (1, 1+1, 1+1+1... 1-1, 1-1-1...)? Thank you!!!!! – 2017-01-21
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    Yes, and a more appropriate way to write the above set is : $(1-1, 1, -1, 1+1, -1-1, 1+1+1, -1-1-1,...)$ just to ensure that all integers are present in the set. Because if you write $(1, 1+1, 1+1+1... 1-1, 1-1-1...)$, then the negative integers are actually never reached. Just a notation though. – 2017-01-21
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    Hmm, a second interpretation of "is generated by $1$ or by $-1$" comes to mind. At first I interpreted it as "and", as mentioned in [this](http://math.stackexchange.com/questions/2107259/mathbbz-is-a-cyclic-group-but-who-generatez-0-zero-element#comment4333264_2107265) comment. But now I think it could also mean $\mathbb Z=\langle \{-1,1\}\rangle$. If it is the second interpretation, I can see the current comments and answer being relevant, otherwise I don't. – 2017-01-21

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Every element in $\mathbb{Z}$ can be obtained from a generator $a$ by the groups law, which is addition. However, we also can use the inverse, which is $-a$. This way we obtain every integer for $a=1$ or $a=-1$. In particular we obtain $0=1+(-1)$. For the definition of a "generator" for a group and examples, see here. Often the group law is written $a\circ b$, but for abelian groups the conventions is usually to write addition, i.e., $a\circ b=a+b$, so that $a^{-1}$ is $-a$, and the neutral element $e$ is $0$. The multiplicative version of your question is, how to obtain $e$: as $a\circ a^{-1}$.

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    To be honest I don't see how this answers the question. The OP knows that the group is generated by $1$ and $-1$, this means that $\mathbb Z=\{1^k\colon k\in \mathbb Z\}$ and $\mathbb Z=\{(-1)^k\colon k\in \mathbb Z\}$ (with a proper definition of power). What could "who generates $0$?" mean except for what power $k$ yields $0$? – 2017-01-21
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    But this is the same: $e=a\circ a^{-1}=a^0$. – 2017-01-21
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    The exponent in the generator isomorphism indicates composition by the group operation with the generator, not multiplication, unless the group operation is explicitly defined as multiplication. – 2017-01-21