Fraleigh Exercise 4.39

Prove that a nonempty set G together with a binary operation * on G such that ax=b and ya=b have [unique] solutions for all a and b in G is a group.

(1) The binary operation is associative.

(2) Fix a in G. Then the equation ax=a has a unique solution, call it e. Then ae=a. Now pick an arbitrary b in G. Then the equation ya=b has a unique solution, call it c. Then ca=b. Multiplying ae=a by c on the left, cae=ca. Thus, be=b. Therefore, e is a right identity on G.

(3) Again pick an arbitrary b in G. The equation bx=e has a unique solution, call it b'. Therefore, every element b as a right inverse in G.

This suffices to show that G is a group.