In this statement and example
The set of all nonnegative integers (including 0) under addition is not a group. There is an identity element 0, but no inverse for 2
This is my confusion. Isn't 0, under addition, an inverse for all nonnegative integers? That is
a * a' = a'*a = a + a' = a' + a
Let $a = 2$ and $a' = 0$ and then $2 + 0 = 0 + 2$
Which is also doing the identity element (does this have to be unique for a group G to be a group?)