Prove that any non empty set G with a binary operation satisfying the following requirements is a group:
a) if x, y in G then x.y in G
b) for all x,y,z in G, (x.y).z = x.(y.z)
c) for all a,b in G, the equations x.a =b and a.y =b both have solutions.
Hint: 1st show that if a in G then a has a right and left identity and be sure to show they are the same.
2ed show that the identity in step #1 works for any b in G
Note: 1st and 2ed steps can be reversed
3ed show that for any a in G a has a left and right inverse and they are the same.
Given that for all a,b in G, the equations x.a =b and a.y =b both have solutions, hence we should have a solution for the equation x.a=a too.
This implies that there is an element 'e' (the solution for the above equation for x=e) such that e.a=a, ...
This solution shows how to prove that any non empty set G with a binary operation, satisfying the following requirements is a group.