**CYCLIC GROUP**

**Also known as Monogenic Group**

**A cyclic group**is the group which can be generated by a single element.

Symbolically
a cyclic group is represented as,

G = {a

^{n }: n ∈ I }
Here group ‘G‘ is generated by element ‘ a ‘.

Here element ‘a’ is called

**group generator.**
Here ‘n’ is called finite

**group order**.
Here ‘I’ is called

**identity element.**
A cyclic
group is

**Abelian**.
Cyclic group
is also called

**monogenic**, because it is generated by a single element.
For example:

G = { 1, -1,
i , -i } is cyclic.

We can write
it as,

G = { i

^{1 }, i^{2}, i^{3}, i^{4}}
Here,

‘

**G**’ is a**cyclic group**and ‘**i’**is a**generator.**
Also can be
written as,

G = { ( -i

^{1 }) , ( -i^{2 }) , ( -i^{3 }) , ( -i^{4 }) }
Here,

‘

**G**’ is a**cyclic group**and ‘**– i**‘ is a**generator.**

**Problems based on above study:**

**Prob.**Show that G = { 1, -1, i, -i } is a cyclic group under multiplicative where i is the imaginary quality such that i

^{2 }=-1.

**SEE ALSO:**