The book:
The notation ka in additive notation does not represent a product of k and a but, rather, a sum $ka=a+a+a+\cdots+a$ with k terms
I confused about the above definition, because we say that $(-1)$ is generator for integers,but we need negative $k$ values to obtain positive integers. If $ka$ is not product, then how can we cancel out the negative sign for getting positive integers ?
I am beginner at abstract algebra and not native speaker. Maybe my question seems stupid, but I really did not understand what it is wanted to mean by saying "The notation ka in additive notation does not represent a product of k and a",because if it is not product, then we cannot obtain the positives
To gather up the several good comments, etc.:
First, there are at least two definitions/characterizations of "subgroup $H$ of group $G$ generated by a subset $S$ of $G$". The slightly fancy one, but whose utility we can eventually see, is to take $H$ to be the intersection of all subgroups of $G$ containing $S$. (And prove that arbitrary intersections of subgroups are, again, subgroups.) This characterization skirts the issue in the question.
Second, define the subgroup-generated-by $S$ to be the collection of all "words" in $S$... and in their inverses! Otherwise, it won't be a subgroup.
In the simplest case of the latter, where $S=\{a\}$ as in the original question, it's not just $a+\ldots+a$ expressions, but, also, $(-a)+\ldots+(-a)$ expressions (and $0$, the "empty sum", by (reasonable) convention).
The issues about $k\cdot a$ and $(-k)\cdot a$ and so on are about notation, not so much about mathematical facts. Yes, there is indeed some checking to be done to be sure that everything is well-defined, and that presumed properties really do hold (depending, of course, on the choice of logical development of these basics...) Yes, this checking is not completely trivial, although it's easy. :)