Okay so lets say G is my finite set with G = { g1, g2, g3,... gn } and we define an operation on G with * : G x G --> G so that we have an abelian group with (G,*)

So for every gi in G we have an inverse gj in G, so that gi * gj = 0G (neutral element)

So cant I just switch around my elements in my finite set (g1*g2*g3*...*gn) so that I always have one element together with its inverse element? it should be possible because it is assoziative and commutative (because its an abelian group).

So both elements as a pair equal the neutral element 0G. This means that I am multiplying 0G with 0G over and over. The neutral element in multiplication is "1". So I do 1*1*1*1.

So isnt g1 * g2 * g3,..., * gn = 1?