How do you prove G is abelian?

Excuse me, could you elaborate on how one might prove that a group G is abelian? I understand that an abelian group is one in which the order of multiplication does not matter, meaning for any two elements a and b in G, the product ab equals ba. But I'm curious about the specific steps or properties one should look for to conclusively demonstrate that G possesses this characteristic. Would it involve examining the group's operation table, verifying certain algebraic identities, or perhaps analyzing the structure of the group's elements? I'm seeking a clear and concise method to approach this question.