Modular Addition of 10 - Abelian Group

This section provides an Abelian Group using the modular arithmetic addition of 10 (integer addition operation followed by a modular reduction of 10).

An Abelian Group can also be defined on at finite number of integers with the help of modular operation.

For example, here is an Abelian Group of 10 integers:

• The set of elements is the set of integers from 0, 1, 2, ..., to 9.
• The binary operation is the integer addition operation followed by a modular reduction of 10 (also called modular arithmetic addition of 10).
• The identity element is the integer 0.

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

```(6 + 7) mod 10 = 3                The "closure" condition
(6 + 7) mod 10 = (7 + 6) mod 10   The "commutativity" condition
((6 + 7) mod 10) + 8) mod 10 = (6 + (7 + 8) mod 10) mod 10
The "associativity" condition
(3 + 0) mod 10 = 3                The "identity" condition
(3 + 7) mod 10 = 0                The "symmetry" condition
```

In fact the above example can be generalized to any positive integer i to define an Abelian Group with i integers:

• The set of elements is the set of integers from 0, 1, 2, ..., to i-1.
• The binary operation is the integer addition operation followed by a modular reduction of i (also called modular arithmetic addition of i).
• The identity element is the integer 0.

We can call the above as Integer Additive Group of Order i, and denote it as G(i,+).