Modular Multiplication of 10 - Not Abelian Group

This section demonstrates that the modular arithmetic multiplication of 10 (integer multiplication operation followed by a modular reduction of 10) can not define an Abelian Group.

In the previous tutorial, we demonstrated that the first 10 non-negative integer can be defined as an Abelian Group with modular arithmetic addition of 10.

Now we want to see if we can defined as an Abelian Group with modular arithmetic multiplication of 10, G(10,*).

The binary operation is the integer multiplication operation followed by a modular reduction of 10.

The "identity" condition is easy to satisfy. Integer 1 should be the identity element. Because I * 1 mode 10 = I is true for any integer.

Now let's check the "symmetry" condition:

If I = 1, then -I = 1, because 1 * 1 mod 10 = 1.
If I = 2, then there is no -I that 2 * (-I) mod 10 = 1. 
If I = 3, then -I = 7, because 3 * 7 mod 10 = 1.
If I = 4, then there is no -I that 4 * (-I) mod 10 = 1
If I = 5, then there is no -I that 5 * (-I) mod 10 = 1
If I = 6, then there is no -I that 7 * (-I) mod 10 = 1
If I = 7, then -I = 3, because 7 * 3 mod 10 = 1.
If I = 8, then there is no -I that 8 * (-I) mod 10 = 1
If I = 9, then -I = 9, because 9 * 9 mod 10 = 1.

As you can see, there are a number of elements that have no symmetric elements.

So G(10,*) is not an Abelian Group.

Last update: 2019.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

 What Is Abelian Group

 Niels Henrik Abel and Abelian Group

 Multiplicative Notation of Abelian Group

 Additive Notation of Abelian Group

 Modular Addition of 10 - Abelian Group

Modular Multiplication of 10 - Not Abelian Group

 Modular Multiplication of 11 - Abelian Group

 Abelian Group on Elliptic Curve

 Discrete Logarithm Problem (DLP)

 Finite Fields

 Generators and Cyclic Subgroups

 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