What Is Subgroup Generator in Abelian Group

This section describes subgroup generator in a Abelian Group. A subgroup generator is an element in an Abelian Group that can be used to generator a subgroup using a series of scalar multiplication operations.

What Is Subgroup Generator in Abelian Group? A subgroup generator is an element in an Abelian Group that can be used to generator a subgroup using a series of scalar multiplication operations as defined below in additive notation:

```Given an element P in an Abelian Group G,
if P, 2P, 3P, ..., is a subgroup S,
P is called the generator of subgroup S.
```

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 Generator Example 1.1 - Integer 3 is the generator of the subgroup of 0, 3, 6, 9, 12, 15, 18, 21. because:

```P = 3
2P = 6
3P = 9
4P = 12
5P = 15
6P = 18
7P = 21
8P = 0
9P = 3
...
```

Subgroup Generator Example 1.2 - Integer 4 is the generator of the subgroup of 0, 4, 8, 12, 16, 20, because:

```P = 4
2P = 8
3P = 12
4P = 16
5P = 20
6P = 0
7P = 4
...
```

Group Example 2 - Let's look at the multiplicative Abelian group of the Binary Field GF(32)/(x2+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 Generator Example 2.1 - Polynomial 1 is a generator of the subgroup of 0, 1, 2, because:

```P = 1
2P = 2
3P = 0
4P = 1
5P = 2
6P = 0
...
```

Subgroup Generator Example 2.2 - Polynomial x is a generator of the subgroup of 0, x, 2x, because:

```P = x
2P = 2x
3P = 0
4P = x
5P = 2x
6P = 0
...
```

Subgroup Generator Example 2.3 - Polynomial x+1 is a generator of the subgroup of 0, x+1, 2x+2, because:

```P = x+1
2P = 2x+2
3P = 0
4P = x+1
5P = 2x+2
6P = 0
...
```
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