2. Let G be a group , H is called the cyclic subgroup of G
generated by a , if H = 𝑎 𝑛 /𝑛𝜖𝑍
A group G is cyclic if there exists an element
a 𝜖 𝐺 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 < a > = G
Cyclic Groups
3. ( Z, + ) is a cyclic group.
( R,+) is not a cyclic group.
( nz,+) is a cyclic group.
G= { 1,i, - 1,-i} is a cyclic group.
Examples
4. A cyclic group can have more than one generator.
1 and -1 are generators of ( z, + )
Generators
5. Let G be a group. Let a 𝜖 G. The least positive integer
n such that 𝑎 𝑛 = e is called the order of a.
Order of an Element
6. In ( 𝐶∗ , . ) i , is an element of order 4 , since 𝑖4 = 1.
In ( 𝑅∗ , .) , -1 is an element of order 2,since −1 2=1.
Example
