How to Divide: Lagrange’s Theorem, Cosets, and an Application to Number Theory

Chapter 2. How το Divide: Lagrange’s T heorem, Cosets, and an A pplication to Number T heory

Our goal today is to introduce subgroups and show that the order of a subgroup divides the order of the group. This raises a basic question in arithmetic: Given two positive numbers m and n, how do we verify that m divides n (written m|n)? Two methods are already available from elemen­ tary school:

(1) The Euclidean Algorithm says that n = qm-\-r for a suitable quotient q > 0 and remainder r with 0 < r < m, so one need merely check r = 0.