# Group cyclic

In a **group cyclic** (G,*) every element x∈G is a power of an element x=a^{n} where a∈G is an element of G called **generator of G** and n is an integer. $$ x=a^n \ \ \ \ \forall \ x \ \in G \ , \ a \in G \ , \ n \in Z$$

**Properties of cyclic groups**

- Every cyclic group is abelian group.
- Every subgroup of a cyclic group is itself a cyclic group
- In a finite cyclic group G of order n, each element m can be a generator of G if and only if GCD (n, m) = 1. If n and m are coprime numbers, then m is a generator of G.

**Example**

The set of integers Z is a group (Z,+) with respect to addition.

$$ (Z,+) $$

The additive group (Z, +) is a cyclic group with **generator a = 1** since every element x ∈ Z is x = n·a where n ∈ Z is an integer.

$$ x = n \cdot 1 $$

For example, x=4 is generated by 4=1+1+1+1

$$ 4 = 4 \cdot a = 1 + 1 + 1 + 1 $$

The element a = 1 is a generator of G because it generates every integer of Z.

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