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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 $1$.

A group generated by one element is called cyclic because the elements of the generated subgroup can be viewed as cycling through powers (or multiples) of the generator. In finite cases like $\mathbb{Z}/n\mathbb{Z}$, the sequence of multiples of the generator repeats periodically, forming a cycle.

Conditions: The group is generated by a single element.; In the finite case, the modulus $n$ determines the length of the cycle.

A group generated by one element is called cyclic because the elements of the generated subgroup can be viewed as cycling through powers (or multiples) of the generator. In finite cases like $\mathbb{Z}/n\mathbb{Z}$, the sequence of multiples of the generator repeats periodically, forming a cycle.

Conditions: The group is generated by a single element.; In the finite case, the modulus $n$ determines the length of the cycle.