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I am trying to understand cyclic, and need to know what the cyclic subgroup of integers (Z) generated by -1 under + ?

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Keep adding $-1$ to itself; what do you get? Then don't forget that subgroups include the inverses; what is the additive inverse of $-1$, and what do you get when you add that to itself? That is the subgroup of $\mathbb{Z}$ you get. (Note that subgroups don't have to be proper.)

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The subgroup would be denoted $\langle -1 \rangle$ and the set of elements of the group is $$\{ \dots , (-1)^{-2}, (-1)^{-1}, (-1)^0, (-1)^1, (-1)^2, \dots \}$$ where $(-1)^n=\underbrace{(-1)+\cdots +(-1)}_\text{$n$-many}$ for $n>0$.

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