Element Generated Subgroup Is Cyclic

This section describes the fact that all subgroups generated from elements in finite Abelian groups are cyclic groups.

If we take any element P from an finite Abelian group and generate a subgroup S, we know that S is a cyclic group. Because the way S is generated matches perfectly the definition of cyclic group.

Cyclic Subgroup Example 1 - Here is a cyclic subgroup of integers: 0, 3, 6, 9, 12, 15, 18, 21, generated from integer 3 in the parent group of 0, 1, 2, 3, ..., 23 using the addition operation with modular reduction of 24.

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

Cyclic Subgroup Example 2 - Here is a cyclic subgroup of polynomials: 0, x+1, 2x+2, generated from polynomial x+1 in the parent group of 0, 1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2 using the algebraic addition operation with modular reduction of 3 on coefficients.

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

Cyclic Subgroup Example 3 - Here is a cyclic group of integers: 1, 3, 4, 5, 9, generated from integer 3 in the parent group of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, using the multiplication operation with modular reduction of 11.

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