If a group has order p and p is a prime, then we call that group to be a group of prime order. A group of prime order has a nice description, and they can be characterized as follows:

- A group of prime order is cyclic, so abelian.
- A group of prime order p is isomorphic to the quotient group Z/pZ.

In this post, we will learn about groups of prime orders with their properties.

Table of Contents

## Group of prime order is cyclic

**Theorem:** A group of order p where p is a prime number is cyclic.

**Proof:**

Let G be a group order p. Since p is a prime number and |G|=p>1, there exists a non-identity element a ∈ G. Let H = <a> be the cyclic subgroup of G generated by the element a. Thus we have

order of a = order of H = |H| > 1.

Since G is a finite group of order p and H is a subgroup of G, by Lagrange’s theorem we deduce that |H| is a divisor of |G|.

As |G|=p and p is a prime, we obtain that the order of H is either 1 or p. As |H| > 1, we conclude that |H|=p. As a result, we have that

H is a cyclic subgroup of G and |H|=|G|.

⇒ H = G.

So the group G is a cyclic group of order p. In other words, a group of prime order is always cyclic.

**Read These:** First Isomorphism Theorem of Groups

Second Isomorphism Theorem | Third Isomorphism Theorem

Conjugacy Relation | Orbit Stabiliser Theorem

**Remark:** A cyclic group is not necessarily of prime order. Note that (Z_{4}, +) is a cyclic group of order 4, but it is not of prime order.

**Also Read:**

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

## Group of prime order is abelian

**Theorem:** A group of order p where p is a prime number is abelian.

**Proof:**

From the above theorem, we know that a group of prime order is cyclic. As a cyclic group is always abelian, we deduce that a group of prime order is an abelian (or commutative) group.

You Can Read: Left and Right Cosets | Normal Group

## FAQs

**Q1: Is a group of prime order cyclic?**

Answer: Yes, a group of prime order (say p) is cyclic as the group contains an element of order p.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.