TITLE: Why is the set of integers with the operation of addition considered a cyclic group?
QUESTION [12 upvotes]: The first sentence in the Wikipedia article entitled "Cyclic Groups" states that "In algebra, a cyclic group is a group that is generated by a single element".
How is this consistent with addition on the set of integers being considered a cyclic group.  What would be the single element that generates all the integers.?
Please don't tell me it is the element 1 :)

REPLY [14 votes]: It is the element $1$ (or the element $-1$).
Why? The subgroup $\langle g\rangle\subset G$ generated by an element $G$ is defined as the smallest subgroup to contain $g$. Since $1$ is in $\langle 1\rangle$ (in $\mathbb Z$) and any subgroup is closed under inverses, $-1$ is also in $\langle 1\rangle$ (since it is the inverse of $1$). Clearly all positive integers are there, and so are their inverses. You get $0$ by the identity group axiom (that the additive identity must be an element of your additive subgroup).