What Is Order of Element

This section describes the order of a given element in a finite Abelian Group, which is defined as the least positive integer n, such that the scalar multiplication of n and P is 0, where 0 is the identity element.

What Is the Order of an Element in a Finite Abelian Group? The order of an element P in a finite Abelian Group is the least positive integer n, such that the scalar multiplication of n and P is 0, where 0 is the identity element.

The above definition is based on the following fact:

For any given element P in a finite Abelian Group, a positive integer m exists, such that the scalar multiplication of m and P is 0. Here is the proof:

```Given element P in an finite Abelian Group G
a positive integer m exists, such that mP = 0.

Proof:
Since G is finite, we must have 2 distinct integers i and j, such that
iP = jP
Otherwise, we will have infinite elements in G.

If j > i, we found m: m = j-i and m>0:
mP = (j-i)P = 0, because iP = jP

If i > j, we found m: m = i-j and m>0:
mP = (i-j)P = 0, because iP = jP
```

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

Order of Element Example 1.1 - The order of integer 3 is 8, because:

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

Order of Element Example 1.2 - The order of integer 4 is 6, because:

```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 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.

Order of Element Example 2.1 - The order of Polynomial 1 is 3:

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

Order of Element Example 2.2 - The order of Polynomial x is 3:

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

Order of Element Example 2.3 - The order of Polynomial x+1 is 3:

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