Lagrange's Theory of Finite Groups and the Euler-Fermat Theorem.

In my Group Theory post, I stated that once a Set of mathematical elements and a mathematical operation have been proven to satisfy some basic properties, all theories, proofs and facts from the mathematical discipline of Group Theory, would automatically apply.

 Lagrange's Theory of Finite Groups states that any group can only be divided up into subgroups of the same size.

Euler-Fermat's Theorem is an application of this, that says that any number (coprime to n), rasied to a special power (called the totient of n) will give 1 mod n.

The coprime and totient properties of the second theory are a consequence of the structure of the Multiplicative Group of Integers Modulo n.