Group of prime order

Verbal definition
A group of prime order, or cyclic group of prime order, is any of the following equivalent things:


 * 1) It is a cyclic group whose order is a prime number.
 * 2) it is isomorphic to the quotient of the group of integers by a subgroup generated by a prime number.
 * 3) It is a simple abelian group.
 * 4) It is the additive group of a finite prime field (note that we have to say finite because "prime field" also includes the field of rational numbers, the prime field of characteristic zero).

If $$p$$ denotes the order of the group, then the cyclic group of order $$p$$ is denoted $$C_p$$, $$\mathbb{Z}/p\mathbb{Z}$$, $$\mathbb{Z}_p$$ or $$\mathbb{F}_p$$.

Properties
Groups of prime order are cyclic; hence they are also Abelian, nilpotent and solvable. Also, a group of prime order is simple.

The fact that any simple Abelian group is cyclic of prime order is extremely useful in the study of composition series for groups, particularly for solvable groups.

Endomorphisms
Any endomorphism of the additive group of a prime field is described completely by where it sends the generator. If the generator, say $$1$$, goes to $$a$$, then any $$x$$ must go to $$ax$$.

Thus, the endomorphisms are parametrized by elements of the prime field. A composition of two endomorphisms is given by their product via the multiplication in the prime field.

In symbols, any element $$a \in \mathbb{F}_p$$ gives the endomorphism:

$$x \mapsto ax$$

The composite of the endomorphisms induced by $$a$$ and $$b$$ is the endomorphism induced by their product $$ab \in \mathbb{F}_p$$.

Automorphisms
The automorphism group is given by those endomorphisms that are given by multiplication by nonzero elements. Thus, the automorphism group is the multiplicative group of nonzero elements in $$\mathbb{F}_p$$. It turns out that this automorphism group is a cyclic group. This follows from the more general fact that any finite subgroup of the multiplicative group of a field is cyclic.

Upto conjugacy
Since the group is Abelian, every element forms its own conjugacy class.

Upto automorphism
The automorphism group acts transitively on non-identity elements: any non-identity element can be sent to another non-identity element via an automorphism (namely, via multiplication by a suitable number). Thus, there are two classes of elements upto automorphism: the identity element (denoted 0 in the additive notation) and the class of non-identity elements, that has size $$p-1$$.

Subgroups
There are only two subgroups: the trivial subgroup and the whole group. Any nontrivial element generates the whole group.

Quotients
There are only two quotients: the trivial quotient and the whole group.

Occurrence as a subgroup
The group of order $$p$$, for any prime $$p$$, occurs as a subgroup (in possibly many ways) in any group whose order is a multiple of $$p$$. This result is known as Cauchy's theorem. It can be proved directly or by combining Sylow's theorem and the fact that a group of prime power order contains subgroups of every order dividing the group's order.

Occurrence as a normal subgroup
The cyclic group of prime order may occur as a normal subgroup in some groups, though this is a rare occurrence. When the prime is very small, the chances are high that a normal subgroup of that order is actually a central subgroup: it lies inside the center. In fact, a normal subgroup whose order is the smallest prime dividing the order of the group, must be central.

One case where this happens is for the general linear group over a field whose order is a prime (or prime power) $$q$$, where $$p \mid q - 1$$. Here, the cyclic group of order $$p$$ occurs as a subgroup of the group of scalar matrices in the general linear group, which is cyclic of order $$q - 1$$.

Occurrence as a quotient group
The cyclic group of prime order often occurs as a quotient. In fact, any normal subgroup of prime index must have as its quotient the cyclic group of prime order. The smaller the prime, the larger the probability that a subgroup of that prime index is normal. For instance, any subgroup of index two is normal and any subgroup of least prime index is normal.