Some Basic Algebra with Exercises: Part 1.

This is what is called the Index Theorem in group theory. From this directly follows the very often used Lagrange Theorem:

One important consequence of this is that a group which order is prime must be a cyclic group. Moreover, each element different from the neutral element is a generator of the group. In more details:

Cyclic groups have an important characterization of its order:

Exercise:

Solution:

Thank you for reading!