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=<a^r> 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, +).

Answer:

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.

Answer:

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.

Answer:

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.

Question	Answer
Is Z cyclic?	As any integer can be written as n= 1+1+…+1 (n times), we see that (Z, +) is generated by 1. Hence, Z is a cyclic group.
Give an example of a non-cyclic group of order 4.	The group Z2 × Z2 has order 4 and it has no element of order 4. So Z2 × Z2 is a noncyclic group of order 4.
Give an example of a cyclic group of order 4.	Z4 is a cyclic group of order 4 generated by $\overline{1}$.

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