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:

• The element set has 8 integers: 0, 3, 6, 9, 12, 15, 18, 21.
• The operation is still the same addition operation with modular reduction of 24.
• The identity element is the same: 0.

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

• The element set has 6 integers: 0, 4, 8, 12, 16, 20.
• The operation is still the same addition operation with modular reduction of 24.
• The identity element is the same: 0.

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

• The element set has 9 binary polynomials: 0, 1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2.
• The addition operation is the normal algebra addition with coefficients reduced by modulo 3.
• The identity element is the binary polynomial: 0.

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

• The element set has 3 binary polynomials: 0, 1, 2.
• The addition operation is the normal algebra addition with coefficients reduced by modulo 3.
• The identity element is the binary polynomial: 0.

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 element set has 3 binary polynomials: 0, x, 2x.
• The addition operation is the normal algebra addition with coefficients reduced by modulo 3.
• The identity element is the binary polynomial: 0.

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.