Order of Subgroup and Lagrange Theorem

This section describes Lagrange Theorem which states that the order of any subgroup in an finite Abelian group divides the order of the parent group.

What Is Lagrange Theorem? Lagrange Theorem states that the order of any subgroup in an finite Abelian group divides the order of the parent group.

Lagrange Theorem can be expressed as:

```Given an finite Abelian group G with an order of l,
and subgroup S with an order of n, there is an integer factor x exists,
such that:
n*x = l
```

The factor x in the above expression is also called the cofactor of subgroup S.

The proof of Lagrange Theorem is given in "Lagrange's theorem (group theory)" at wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory).

Group Example 1 - Here is the additive Abelian group of integers: 0, 1, 2, ..., 23 and the addition operation with modular reduction of 24. The order of the group is 24.

Order of Subgroup Example 1.1 - The order of the subgroup generated by 3 is 8, which divides 24 as 8*3 = 24:

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

Order of Subgroup Example 1.2 - The order of the subgroup generated by 4 is 6, which divides 24 as 6*4 = 24:

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

Group Example 2 - Let's look at the multiplicative Abelian group of the Binary Field GF(32)/(x2+1). The order of the group is 9.

Order of Subgroup Example 2.1 - The order of the subgroup generated by Polynomial 1 is 3, which divides 9 as 3*3 = 9:

```P = 1
2P = 2
3P = 0
```

Order of Subgroup Example 2.2 - The order of the subgroup generated by Polynomial x is 3, which divides 9 as 3*3 = 9:

```P = x
2P = 2x
3P = 0
```

Order of Subgroup Example 2.3 - The order of the subgroup generated by Polynomial x+1 is 3, which divides 9 as 3*3 = 9:

```P = x+1
2P = 2x+2
3P = 0
```