Order of Subgroup and Lagrange Theorem

This section describes Lagrange Theorem which states that the order of any subgroup in an finite Abelian group divides the order of the parent group.

What Is Lagrange Theorem? Lagrange Theorem states that the order of any subgroup in an finite Abelian group divides the order of the parent group.

Lagrange Theorem can be expressed as:

Given an finite Abelian group G with an order of l, 
and subgroup S with an order of n, there is an integer factor x exists, 
such that:
   n*x = l

The factor x in the above expression is also called the cofactor of subgroup S.

The proof of Lagrange Theorem is given in "Lagrange's theorem (group theory)" at wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory).

Group Example 1 - Here is the additive Abelian group of integers: 0, 1, 2, ..., 23 and the addition operation with modular reduction of 24. The order of the group is 24.

Order of Subgroup Example 1.1 - The order of the subgroup generated by 3 is 8, which divides 24 as 8*3 = 24:

 
P = 3
2P = 6
3P = 9
4P = 12
5P = 15
6P = 18
7P = 21
8P = 0

Order of Subgroup Example 1.2 - The order of the subgroup generated by 4 is 6, which divides 24 as 6*4 = 24:

 
P = 4
2P = 8
3P = 12
4P = 16
5P = 20
6P = 0
...

Group Example 2 - Let's look at the multiplicative Abelian group of the Binary Field GF(3^2)/(x^2+1). The order of the group is 9.

Order of Subgroup Example 2.1 - The order of the subgroup generated by Polynomial 1 is 3, which divides 9 as 3*3 = 9:

 
P = 1
2P = 2
3P = 0

Order of Subgroup Example 2.2 - The order of the subgroup generated by Polynomial x is 3, which divides 9 as 3*3 = 9:

P = x
2P = 2x
3P = 0

Order of Subgroup Example 2.3 - The order of the subgroup generated by Polynomial x+1 is 3, which divides 9 as 3*3 = 9:

P = x+1
2P = 2x+2
3P = 0

Last update: 2019.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

Generators and Cyclic Subgroups

 What Is Subgroup in Abelian Group

 What Is Subgroup Generator in Abelian Group

 What Is Order of Element

 Every Element Is Subgroup Generator

Order of Subgroup and Lagrange Theorem

 What Is Cyclic Group

 Element Generated Subgroup Is Cyclic

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB