# Classification of natural numbers for which the multiplicative group is cyclic

This article states and (possibly) proves a fact that is true for certain kinds of odd-order groups. The analogous statement is not in general true for groups of even order.

View other such facts for finite groups

## Contents

## Statement

Let be a natural number greater than . Then, the Multiplicative group modulo n (?) is a cyclic group if and only if is or of the form or for some odd prime .

### Caution

Note that the multiplicative group modulo is a totally different concept from the multiplicative group of a finite field of order . The former arises as the multiplicative group of a ring that is far from a field, while the latter is the multiplicative group of a finite field. When , the notions coincide.

Also, the multiplicative group of a finite field is *always* cyclic, even when the prime is .

## Related facts

- Multiplicative group of a prime field is cyclic
- Multiplicative group of a finite field is cyclic
- Multiplicative group of a field implies every finite subgroup is cyclic
- Classification of fields whose multiplicative group is locally cyclic

## Examples

For examples of prime numbers, refer Multiplicative group of a prime field is cyclic#Examples.

Here, we discuss examples of prime powers.

In this case, the multiplicative group is , and is cyclic on .

In this case, the multiplicative group is , and it has two possible generators: and .