Which its the subgroup generated by -1 respect to +?

Question

I have the question:
Which its the cyclic subgroup of $\mathbb{Z}$ generated by -1 respect to +? I know that its all $\mathbb{Z}$ but I don´t understad why?
Help, please.

Answer 1

Hints:

The subgroup generated $-1$ is all integer multiples of $-1.$ Do you see why that's the case? If you cannot see it, please read the definition again.

Answer 2

To my taste, a more positive way to think about "subgroup generated by subset $S$ of group $G$" is not to declare that it is the collection of all expressions involving elements of $S$ (and their inverses!), but, rather, that it is the "smallest subgroup of $G$ containing the set $S$". That there exists a smallest such subgroup, and that it is unique, is by observing that the intersection of all subgroups containing $S$ is a subgroup, contains $S$, and is a subgroup of any subgroup containing $S$.

So, in the case at hand, since "the subgroup generated by $-1$" (or, similarly, by $1$) first of all needs to be a subgroup, it has to contain additive inverses... so if it contains $-1$ it must contain $+1$, and vice versa. "Rules" about allowable expressions are secondary.