Element Generated Subgroup Is Cyclic

This section describes the fact that all subgroups generated from elements in finite Abelian groups are cyclic groups.

If we take any element P from an finite Abelian group and generate a subgroup S, we know that S is a cyclic group. Because the way S is generated matches perfectly the definition of cyclic group.

Cyclic Subgroup Example 1 - Here is a cyclic subgroup of integers: 0, 3, 6, 9, 12, 15, 18, 21, generated from integer 3 in the parent group of 0, 1, 2, 3, ..., 23 using the addition operation with modular reduction of 24.

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

Cyclic Subgroup Example 2 - Here is a cyclic subgroup of polynomials: 0, x+1, 2x+2, generated from polynomial x+1 in the parent group of 0, 1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2 using the algebraic addition operation with modular reduction of 3 on coefficients.

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

Cyclic Subgroup Example 3 - Here is a cyclic group of integers: 1, 3, 4, 5, 9, generated from integer 3 in the parent group of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, using the multiplication operation with modular reduction of 11.

P = 3
P2 = 9
P3 = 27 = 5, (mod 11)
P4 = 81 = 4, (mod 11)
P5 = 243 = 1, (mod 11)

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