# Group of prime order

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Definition

### Verbal definition in terms of the prime

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 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 $\mathbb{F}_p$.

### Verbal definition independent of the prime

A group of prime order is a nontrivial group satisfying the following equivalent conditions:

1. It has exactly two distinct subgroups: the trivial subgroup and the whole group.
2. It is a simple abelian group.

### Equivalence of definitions

Further information: Equivalence of definitions of group of prime order

## Endomorphisms

### 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. Further information: finite subgroup of multiplicative group of field is cyclic

## Elements

### Up to conjugacy

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

### Up to 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.

## In larger groups

### 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.

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
lattice-determined group property Yes Given two groups $G_1, G_2$ with isomorphic lattices of subgroups, they are either both groups of prime order, or neither is. The explicit condition on the lattice of subgroups is that it has two elements.

## Relation with other properties

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
cyclic group of prime power order cyclic, and prime power order |FULL LIST, MORE INFO
abelian group of prime power order abelian, and prime power order Cyclic group of prime power order|FULL LIST, MORE INFO
group of prime power order order is a power of a prime |FULL LIST, MORE INFO
finite simple group finite, and no proper nontrivial normal subgroup |FULL LIST, MORE INFO
finite cyclic group Cyclic group of prime power order|FULL LIST, MORE INFO