Group cyclic

In a group cyclic (G,*) every element x∈G is a power of an element x=aⁿ 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.