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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 $0$?

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    $\begingroup$ Note that "generates" is context-dependent in mathematics. $1$ generates the group $\mathbb{Z}$ but $1$ does not generate the monoid $\mathbb{Z}$ (as in the latter case, the set of non-negative integers is a strictly smaller monoid containing $1$) $\endgroup$
    – Brian Moehring
    Commented Apr 26, 2020 at 6:09

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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 $\mathbb Z$.

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  • $\begingroup$ so you mean to say $g^n=h$ is equivalent to $(g^{-1})^{-n}=h$ where $g$ and $g^{-1}$ are generators and $h$ is some element in the Group. $\endgroup$
    – Umesh shankar
    Commented Apr 26, 2020 at 6:11
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    $\begingroup$ @Umeshshankar Yes. The laws of exponents apply to groups as well. $\endgroup$
    – Parcly Taxel
    Commented Apr 26, 2020 at 6:12
  • $\begingroup$ ok when we say $g^n=h$ it means $h$ is generated by $n$ copies of $g$ which also means $h$ is generated by $-n$ copies of $g^{-1}$. Does this makes sense? $\endgroup$
    – Umesh shankar
    Commented Apr 26, 2020 at 6:15
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    $\begingroup$ @Umeshshankar Yes. $\endgroup$
    – Parcly Taxel
    Commented Apr 26, 2020 at 6:15
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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 $\langle-1\rangle=\langle1\rangle=\Bbb Z$.

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