In any math textbook, there is always the question: Does $\Bbb N$ contain 0? Some authors say yes, some say no, it’s mostly a matter of taste and utility.
I use $\Bbb Z^+$ to denote the positive integers.
More generally, if $A\subseteq \Bbb R$ is any subset of the real numbers, I use the notation $A^+$ to denote the subset of positive numbers. That is to say,
$$ A^+ = \{x\in A: x > 0\} $$
By this convention, it makes sense to use $\Bbb N=\{0,1,2,\dots\}$ and $\Bbb Z^+=\{1,2,3,\dots\}$.
Throughout the rest of the chapters on group theory, we will assume that $(G,\star)$ is an abstract group.