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To show that the integers $\mathbb{Z}$ under addition are cyclic, demonstrate that $\mathbb{Z} = \langle 1 \rangle$. The subgroup generated by $1$ contains $1$, its inverse $-1$, the identity $0$, and all integer multiples of $1$, which exhausts all integers.

Conditions: The group is $\mathbb{Z}$ with operation $+$.; The candidate generator is $1$.

To show that the integers $\mathbb{Z}$ under addition are cyclic, demonstrate that $\mathbb{Z} = \langle 1 \rangle$. The subgroup generated by $1$ contains $1$, its inverse $-1$, the identity $0$, and all integer multiples of $1$, which exhausts all integers.

Conditions: The group is $\mathbb{Z}$ with operation $+$.; The candidate generator is $1$.