In this statement and example

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> 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
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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?)