Cyclic Group with Examples

In this class, We discuss the Cyclic Group with Examples.

The reader should have prior knowledge of the group. Click Here.

Cyclic Group:

A group [G *] is said to be a cyclic group

if there exists a ∈ G such that every element x ∈ G can be written as a^n

a is called a generator.

Example:

1, w, w^2 are cube roots of unity.

1, w, w^2 form a group with multiplication.

The example is shown in our previous classes.

Here w is the generator.

w^1 = w

w^2 = w^2

w^3 = 1

w generating all the elements in the group.

Example 2:

[{1, -1} *) is a cyclic group?

First, check the group conditions.

Closure is satisfying.

Associativity is satisfying.

The identity element is 1.

Inverse elements exist.

The generator is -1.

-1^1 = -1

-1^2 = 1

Example 3:

[{1, 2, 3, 4} *mod5] is a cyclic group?

The binary operation is multiplication modulo five.

1 * 2 mod five is 2.

The closure property is satisfied.

The associative property is satisfied.

The identity element is 1.

Inverse elements exist.

(1, 1) (2, 3), (4, 4)

generator elements are 2, 3

We can have more than one generator element.