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.
- 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.
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.
⇒ G is a cyclic group of order p.
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.
Group of prime order is abelian
Theorem: A group of order p where p is a prime number is abelian.
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.