The set $G = \{0, 1, 2\}$ under addition modulo 3 is shown to be abelian. Which observation establishes that?

A group is abelian when $a+b=b+a$ for every pair of elements, and a Cayley table symmetric about the leading diagonal says exactly that. Here $1+2$ and $2+1$ both give 0, and every other pair matches in the same way. Being closed, associative and having an identity are needed for a group but say nothing about the order of the operands, and 2 is not its own inverse since $2+2\equiv1$.