Prove the following is a group

[beachbum300]

Homework Statement

Prove that a nonempty set S with an associative binary operation * on S such that a*c=b and d*a=b have solutions in S for all a,b in S is a group.

Homework Equations

The Attempt at a Solution

In order to prove something is a group, I know that I must show associativity of *, existence of identity element and inverse for each element in S.
- We are given that S has an associative binary operation, so that is satisfied.
- In regards to finding the identity element such that a*e=e*a=a for all a in S...this is where I am not sure where to begin. Any suggestions on how to start?

 

[Dick]

Take a and b to be the same element a. Then your premise tells you you can find c and d such that a*c=a and d*a=a. It looks like c and d should be your identity 'e', right? To show that's true, you have to prove c=d, and you have to prove if you pick another element besides a, you get the same identity. It can be done. You just have to keep using the only premise you've got over and over. Similar course for inverses.