What Is Subgroup in Abelian Group

This section describes Subgroups in a Abelian Group. A subgroup in a Abelian Group is a subset of the Abelian Group that itself is an Abelian Group. The subgroup and its parent group are using the same operation.

What Is Subgroup in Abelian Group? A subgroup in an Abelian Group is a subset of the Abelian Group that itself is an Abelian Group. The subgroup and its parent group are using the same operation.

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

Subgroup Example 1.1 - Here is a subgroup in the above group:

Subgroup Example 1.2 - Here is another subgroup in the above group:

Group Example 2 - Here is the multiplicative Abelian group of the Binary Field GF(3^2)/(x^2+1).

Subgroup Example 2.1 - Here is a subgroup in the above group:

The table below defines the addition operation of this subgroup:

   + |    0     1     2 
-----------------------
   0 |    0     1     2
   1 |    1     2     0
   2 |    2     0     1

Subgroup Example 2.2 - Here is another subgroup in the above Abelian Group:

The table below defines the addition operation of this subgroup:

   + |    0     x    2x
-----------------------
   0 |    0     x    2x
   x |    x    2x     0
  2x |   2x     0     x

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