Infinite Cyclic Group of Integers

Question

I am new to group theory. While reading about cyclic groups, according to my understanding, A Cyclic group has a generator that generates all other elements by several copies of it. Now coming to set of integers $Z$ with addition as a binary operation, i read that $Z$ is an infinite cyclic group with generators $1$ and $-1$.

But $1$ cannot generate negative integers no matter how many copies are added and analogously for $-1$ which cannot generate positive integers. So does it mean $1$ generates positive integers and $-1$ generates negative integers? How about generating identity element $0$?

Answer 1

The definition of the group generated by an element or a set of elements allows inverses to be used when composing elements. The generated group always includes the identity, regarded as the composition of nothing.

$-1$ is the inverse of $+1$, so $+1$ alone generates all of $\mathbb Z$. Likewise, $-1$ alone generates all of $\mathbb Z$.

Answer 2

This is one thing that may be a little easier to understand in multiplicative notation. The cyclic group generated by an element $x$ is $\langle x\rangle=\{x^n:n\in\Bbb Z\}$.

In additive it's: $\langle x\rangle=\{n\cdot x:n\in\Bbb Z\}$.

Thus $\langle-1\rangle=\langle1\rangle=\Bbb Z$.