1. The problem statement, all variables and given/known data Prove that a set G, together with a binary operation * on G, satisfying the following three axioms: A1) The binary operation * on G is associative A2) There exists a left identity element e in G such that e*x=x for all x in G A3) For each a in G, there exists a left inverse a' in G such that a'*a=e is a group 2. Relevant equations Our definition of a group: A group is a set G, and a closed binary operation * on G, such that the following axioms are satisfied: G1) * is associative on G G2) There is an element e in G such that for all x in G x*e = e*x = x (existence of an identity element e for *) G3) Corresponding to each a in G there is an element a' in G such that a*a' = a'*a = e (existence of inverse a' of a) 3. The attempt at a solution G1, is given by A1 By A2 we know e*x=x for all x in G. We must show x*e=x for all x in G to finish proving G2. x*e=x, what is to be proved x'*(x*e)=x'*x, we know x' exists from A3, existence of inverse (x'*x)*e=e, using A1, associativity e*e=e, using A3, existence of inverse e=e, therefore x*e=x for x in G proving G2 holds. By A3 we know there exists x' in G such that x'*x=e for all x in G. We must show x*x'=e for all x in G to finish proving G3. x*x'=e, what is to be proved (x'*x)*x'=x'*e, using A1, associativity e*x'=x', using the above theorem x'=x', therefore x*x'=e for x in G proving G3 hold. Therefore the set G, together with a binary operation * on G, satisfying A1, A2 and A3 is a group. 4. My questions Does ending with the e=e statement or x'=x' statement show the original statement holds? Since it ends up with LHS=RHS I think it does but maybe I should try to string statements in the form e*x = ... = x*e although I am not sure why that would be better. I guess it is just what I am used to seeing. Also, I skipped the part about the binary operation being closed for a group. Kind of important, I know. I think (hope) it was indirectly mentioned in the axioms when it was stated * in associative -on- G. Is this correct or is it something I will also need to prove? If so, any ideas? Thank you for taking the time to read all this.