Here is a simple problem from abstract algebra.

Prove that a group with exactly five elements is commutative.

The order of an element must divide the order of the group by Lagrange's Theorem. Since the group has prime order, all elements have either order 1 or 5. Since the identity element is the unique element with order 1, the remaining 4 elements must all have order 5. Any of these elements is thus a generator of the entire group, showing that the group is cyclic. Cyclic implies abelian, completing the proof.

Posted by Jason on 2006-09-18 14:09:27