According to the book I'm reading (Advanced Linear Algebra by Nicholas Loehr, p4-see photo), the integers modulo $n$ with the addition modulo $n$ form a commutative group of size $n$.
But I cannot see how each element has an inverse that satisfies the axiom 4 here. (for every a in the group, there exists $-a$ that leads to $a+(-a)=0$).
I think it would make sense if the integers modulo is symmatric (ex, it is $\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$, with $n$ being $5$).
But the specific example in the book is: $\{0, 1, 2, 3, ..n-1\}$ as you can see.
Is this an error in the book or am I missing something?
As @LordSharktheUnknown describes in the comments . . .
No, it's not an error.
The inverses of the operation $\oplus$ can be delineated with a few examples, like so:
$$\begin{align} 1\oplus (n-1)&=1+(n-1)-n \\ &=0, \\ 2\oplus (n-2)&=2+(n-2)-n\\ &=0, \\ &\vdots \end{align}$$