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Cyclic group

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 = {an : 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 = { i1 , i2 , i3 , i4 }

Here,

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

Also can be written as,

G = { ( -i1 ) , ( -i2 ) , ( -i3 ) , ( -i4 ) }

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=-1.


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