Cyclic Group

A cyclic group is a group in which every element of the group can be obtained by repeatedly applying the group operation to a single element, known as the generator of the group.

If a group (G) has an element (a) such that every element of (G) can be written as a power (or repeated addition) of (a), then (G) is called a cyclic group.

A cyclic group is represented as G = \langle a\rangle where (a) is the generator of the group.

For multiplicative groups, G=\langle a\rangle={a^n \mid n\in\mathbb{Z}}, where (n) is any integer.

For additive groups, G=\langle a\rangle={na \mid n\in\mathbb{Z}}, where (na) denotes repeated addition of (a).

Types of cyclic groups

Finite Cyclic Groups

• The group of integers modulo n, denoted Z_n, is a cyclic group. It consists of the integers {0, 1 , 2, … ,n−1} under addition modulo n. The generator is 1 because every element can be written as 1⋅k (mod  n) for some integer k.

Infinite Cyclic Group

• The group of all integers Z under addition is an infinite cyclic group. The generator can be 1 (or −1), because every integer nnn can be written as 1 ⋅ n or (-1) ⋅ (−n).

Properties

Some of the common properties of cyclic group are:

• Every cyclic group is also an Abelian group.
• If G is a cyclic group with generator g and order n. If m < n, then the order of the element g^m is given by, |g^{m}| = \frac{n}{gcd(m, n)}
• Every subgroup of a cyclic group is cyclic.
• If G is a finite cyclic group with order n, the order of every element in G divides n.
• If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is Φ(d), where Φ(d) is Euler Phi function.
• The order of a cyclic group and the order of its generator is same.

Steps to Determine Cyclic Group

A group is cyclic if there exists at least one element that generates every element of the group.

1. Choose an element of the group.
2. Repeatedly apply the group operation.
3. List all distinct elements obtained.
4. If every group element appears exactly once before the sequence repeats, then the chosen element is a generator.
5. If at least one generator exists, the group is cyclic.

➢Practice: Solved Examples