Question

what is cyclic group

Answer 1

is a group that is generated by a single element

Answer 2

Note :

• Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

• Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

Answer :

Cyclic group : A group G is called a cyclic group , if there exists an element a ∈ G , such that every element x ∈ G can be written as x = aⁿ for some integer n . And the element a is called the generator of G .

Example :

We know that , the set G = { ±1 , ±i } is a group with respect to the multiplication .

Here , 1 is the identity element in G .

If we take the element i ∈ G , then every element in G can be written as iⁿ for some integer n .

1 = i⁴

-1 = i²

i = i¹

-i = i³

Hence , G = { ±1 , ±i } is a cyclic group with the generator i . (Here , -i is also a generator of G) .