"How to prove that a set is a group?" related difficult/challenge quesyion. 1.In proving that a set is a group from definition, we have to show there is an inverse element for each element, do we have to show uniqueness of it? I believe this is unnecessary but people do this all the time. My argument: "For each [itex]a\in G[/itex], there exists a left inverse a' in G such that a'a=e." is enough.