The document defines cyclic subgroups and cyclic groups. A cyclic subgroup is generated by a single element a of a group G. A group G is cyclic if it can be generated by a single element a of G such that the subgroup generated by a is equal to G. Examples of cyclic groups given include (Z,+) and (Zn,+), while (R,+) is not cyclic. The generators of a cyclic group are elements that can generate the entire group. The order of an element a is the least positive integer n such that an = e. Examples of orders of elements in groups are also provided.

1. V.V.Vanniaperumal College for
Women , Virudhunagar.
Modern Algebra
B.Sc Mathematics

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

7. Thank you