Group of Prime Order is Cyclic: Proof

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.

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 (Z₄, +) 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

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