Here is a simple problem from abstract algebra.

Prove that a group with exactly five elements is commutative.

(In reply to re: Counting elements by Tristan)

For all elements a in a group, there exists an element b such that ab = ba = 1. Assume a' and a" are such elements. Then

a' = 1a' = (a"a)a' = a"(aa') = a"1 = a".

I remember when taking a graduate course in algebra that we spent a lot of time proving existence and uniqueness proofs. The existence proofs were generally one or two pages long while the uniqueness proofs were one or two lines long.

Posted by Bractals on 2006-09-19 16:33:19