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.

2006 Nobel Prize in Physics

For their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation

John C. Mather
Goddard Space Flight Center

George F. Smoot
UC-Berkeley

http://nobelprize.org/nobel_prizes/physics/laureates/2006/