Modular Multiplication of 11 - Abelian Group

This section provides an Abelian Group using the modular arithmetic multiplication of 11 (integer multiplication operation followed by a modular reduction of 11).

In the last tutorial, we demonstrated that the modular arithmetic multiplication of 10 can not be used to define an Abelian Group.

But if we change the modular base from 10 to 11, then we can use the modular arithmetic multiplication of 11 to define an Abelian Group.

• The set of elements is the set of integers from 1, 2, ..., to 10.
• The binary operation is the integer multiplication operation followed by a modular reduction of 11 (also called modular arithmetic multiplication of 11).
• The identity element is the integer 1.

You can verify that all 5 Abelian Group conditions are satisfied. For example:

```(6 * 7) mod 11 = 9                The "closure" condition
(6 * 7) mod 11 = (7 * 6) mod 11   The "commutativity" condition
((6 * 7) mod 11) * 8) mod 11 = (6 * (7 * 8) mod 11) mod 11
The "associativity" condition
(3 * 1) mod 11 = 3                The "identity" condition

(1 * 1) mod 11 = 1                The "symmetry" condition
(2 * 6) mod 11 = 1                The "symmetry" condition
(3 * 4) mod 11 = 1                The "symmetry" condition
(4 * 3) mod 11 = 1                The "symmetry" condition
(5 * 9) mod 11 = 1                The "symmetry" condition
(6 * 2) mod 11 = 1                The "symmetry" condition
(7 * 8) mod 11 = 1                The "symmetry" condition
(8 * 7) mod 11 = 1                The "symmetry" condition
(9 * 5) mod 11 = 1                The "symmetry" condition
```

In fact the above example can be generalized to any prime integer p to define an Abelian Group with p-1 integers:

• The set of elements is the set of integers from 1, 2, ..., to p-1.
• The binary operation is the integer multiplication operation followed by a modular reduction of p (also called modular arithmetic multiplication of p).
• The identity element is the integer 1.

We can call the above as Integer Multiplicative Group of Order p, and denote it as G(p,*).

Last update: 2019.