Every Element Is Subgroup Generator

This section describes the fact that every element in an finite Abelian group is a subgroup generator. The order of the generated subgroup is the same as the order of the element.

Another nice fact of an finite Abelian Group is that every element is a subgroup generator, which can be used to generate a subgroup.

To prove this fact, let's express it as:

 
Given an finite group G and any element P, 
the following element set is a subgroup generated by P:
   S = {P, 2P, 3P, ..., (n-1)P, nP=0}
where n is the order of the element P. 

Now we can verify if subgroup S satisfies the 5 rules of Abelian Group or not.

1. Closure:

If I = iP and J = jP are in S, 
Then K = I + J = iP + jP = (i+j)P is in S
   
Proof:
   If i+j <= n, K = (i+j)P is in S by definition
   If i+j > n and i+j < 2n, we have:
      k = n - (i+j), where k > 0 and k < n
      i+j = n+k         
      K = (i+j)P = (n+k)P = nP + kP = kP, since nP = 0
      K = kP must be in S, because k < n

2. Commutativity:

If I = iP and J = jP are in S, 
Then I + J = J + I

Proof:   
   I and J are elements in the parent group, so:
      I + J = J + I, Commutativity of the parent group

3. Associativity:

If I = iP, J = jP and K = kP are in S, 
Then (I + J) + K = I + (J + K)

Proof:   
   I, J and K are elements in the parent group, so:
      (I + J) + K = I + (J + K), Associativity of the parent group

4. Identity Element:

If I = iP is in S, 
Then iP = nP = iP

Proof:
   iP is an element in the parent group and 0 is the identity element, so:
      iP + nP = iP + 0 = iP, where i = 1, 2, ..., n-1

5. Symmetry:

If I = iP is in S, there is j exists, such that:
   iP + jP = 0

Proof:
   i <= n by definition
   If i = n, then
      j = n, and 
      iP + jP = nP + nP = 0 + 0 = 0
   If i < n, then 
      j = n - i, 
      iP + jP = iP + (n-i)P = nP = 0

Note that the order of the subgroup (group size) is the same as the order of the element which generated the subgroup.

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