# Cyclic Group: Definition, Examples, Properties, Orders

A cyclic group is a special type of group generated by a single element. If the generator of a cyclic group is given, then one can write down the whole group. Cyclic groups are also known as monogenous groups. In this article, we will learn about cyclic groups.

## Definition of Cyclic Groups

A group (G, $\circ$) is called a cyclic group if there exists an element a∈G such that G is generated by a. In other words,

G = {an : n ∈ Z}.

The element a is called the generator of G. Mathematically, it is written as follows: G=<a>. If G is an additive cyclic group that is generated by a, then we have G = {na : n ∈ Z}. The following are a few examples of cyclic groups.

• (Z, +) is a cyclic group. Its generators are 1 and -1.
• (Z4, +) is a cyclic group generated by $\bar{1}$. It is also generated by $\bar{3}$.

Non-example of cyclic groups: Klein’s 4-group is a group of order 4. It is not a cyclic group.

## Order of a Cyclic Group

Let (G, $\circ$) be a cyclic group generated by a. The order of group G is equal to the order of the element a in G. In other words, $|G|=|a|$, where $|g|$ denotes the order of the element g. Depending upon whether the group G is finite or infinite, we say G to be a finite cyclic group or an infinite cyclic group.

In the above example, (Z4, +) is a finite cyclic group of order 4, and the group (Z, +) is an infinite cyclic group.

## Properties of Cyclic Groups

• If a cyclic group is generated by a, then it is also generated by a-1.
• Every cyclic group is abelian (commutative).
• If a cyclic group is generated by a, then both the orders of G and a are the same.
• Let G be a finite group of order n. If G is cyclic then there exists an element b in G such that the order of b is n.
• Let G be a finite cyclic group of order n and G=<a>. Then G=<ar> if and only if r<n and gcd(r, n)=1. Thus the number of generators of a finite cyclic group of order n is Φ(n), where Φ is the Euler-Phi function.
• Every subgroup of a cyclic group is also cyclic.
• A cyclic group of prime order has no proper non-trivial subgroup.
• Let G be a cyclic group of order n. Then G has one and only one subgroup of order d for every positive divisor d of n.
• If an infinite cyclic group G is generated by a, then a and a-1 are the only generators of G.

## Problems and Solutions on Cyclic Groups

Question 1: Find all subgroups of the group (Z, +).

We know that (Z, +) is a cyclic group generated by 1. As every subgroup of a cyclic group is also cyclic, we deduce that every subgroup of (Z, +) is cyclic, and they will be generated by different elements of Z.

The cyclic subgroup generated by the integer m is (mZ, +), where mZ={mn: n ∈ Z}. As (mZ,+) = (-mZ, +) we conclude that all the subgroups of the group (Z, +) are given as follows:

{(mZ, +) : m is a non-negative integer}.

Question 2: Prove that (Q, +) is not a cyclic group.

For a contradiction, we assume that (Q, +) is a cyclic group generated by a. Since (Q, +) is an additive group, we can write it as Q={na : n∈ Z}. As $\frac{1}{2}$a ∈ Q and see that $\frac{1}{2}$a does not belong to the set Q={na : n∈ Z}. Thus a cannot a generator of (Q, +).  As a result, the additive group Q is non-cyclic.

Question 3: Prove that (R, +) is not a cyclic group.

If possible, we assume that (R, +) is a cyclic group. Note that (Q, +) is a subgroup of (R, +). Thus being a subgroup of a cyclic group, we obtain that (Q, +) is a cyclic group. Thus we arrive at a contradiction because (Q, +) is non-cyclic follows from the above question. So the additive group of real numbers is not a cyclic group.

Related Topics:

Group Theory: Definition, Examples, Orders, Types, Properties, Applications

Abelian Group: Definition, Properties, Examples

Group Homomorphism

First Isomorphism Theorem

Kernel of a Homomorphism

Normal Subgroup

Center of a Group

## FAQs on Cyclic Groups

Q1: Is the symmetric group S3 cyclic?

Answer: Since S3 is not an abelian group, S3 is not a cyclic group.

Q2: Is the dihedral group D4 cyclic?

Answer: We know that the dihedral group is non-abelian. Thus D4 is not a cyclic group.

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