What Is Cyclic Group

This section describes Cyclic Group, which is a finite Abelian group that can be generated by a single element using the scalar multiplication operation in additive notation (or exponentiation operation in multiplicative notation).

What Is Cyclic Group? A Cyclic Subgroup is a finite Abelian group that can be generated by a single element using the scalar multiplication operation in additive notation (or exponentiation operation in multiplicative notation).

A Cyclic Group can be expressed in additive notation as:

```A finite Abelian Group is a Cyclic Group,
if it's elements can be expressed as:
{P, P+P, P+P+P, ..., P+P+...+P} or
{P, 2P, 3P, ..., nP}, where n is the order of the group
nP = 0 is the identity element
```

A Cyclic Group can be expressed in multiplicative notation as:

```A finite Abelian Group is a Cyclic Group,
if it's elements can be expressed as:
{P, P*P, P*P*P, ..., P*P*...*P} or
{P, P2, P3, ..., Pn}, where n is the order of the group
Pn = 1 is the identity element
```

Cyclic Group Example 1 - Here is a Cyclic group of integers: 0, 3, 6, 9, 12, 15, 18, 21 and the addition operation with modular reduction of 24. Integer 3 is a group generator:

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

Cyclic Group Example 2 - Here is a Cyclic group of polynomials: 0, x+1, 2x+2, and the algebraic addition operation with modular reduction of 3 on coefficients. Polynomial x+1 is a group generator:

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

Cyclic Group Example 3 - Here is a Cyclic group of integers: 1, 3, 4, 5, 9, and the multiplication operation with modular reduction of 11. Integer 3 is a group generator:

```P = 3
P2 = 9
P3 = 27 = 5, (mod 11)
P4 = 81 = 4, (mod 11)
P5 = 243 = 1, (mod 11)
```

Cyclic Group Example 4 - Here is a Cyclic group of integers: 1, 2, 3, 4, 5, 6, and the multiplication operation with modular reduction of 7. Integer 3 is a group generator:

```P = 3
P2 = 9 = 2, (mod 7)
P3 = 27 = 6, (mod 7)
P4 = 81 = 4, (mod 7)
P5 = 243 = 5, (mod 7)
P6 = 729 = 1, (mod 7)
```

If express P6 as P0 in the last example and replace P with Z, we can use the following graph to represent this Cyclic Group: