Yeah buddy, I am just a junior university student. Today, we briefly talked about the groups in class. My professor said that $(G,\ast)$ is a group where $G$ is non-empty set with binary operaiton $\ast$ iff
The binary operation $\ast$ on $G$ is associative
THere exists an identity element $e \in G $ s.t $e \ast x=x=x \ast e$ for all $x \in G$
For each $a \in G$, there exists an inverse $a'$ s.t $a \ast a'=e=a' \ast a$
However, when I read book (first course in abstract algebra), I saw that we dont have to show the identity and inverses from both sides. Showing only from left or right side is enough. Then ,how can I prove that the both sides always gives the same result ? I think that if it were abelian group, then it is okey, but how does it hold for all operations ?
NOTE: I am new in both mathematics and M.S.E, so please give comments if I do somethings wrong here.
