# 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

## Statement

Let $n$ be a natural number greater than $1$. Then, the Multiplicative group modulo n (?) is a cyclic group if and only if $n$ is $2,4$ or of the form $p^k$ or $2p^k$ for some odd prime $p$.

### Caution

Note that the multiplicative group modulo $p^k$ is a totally different concept from the multiplicative group of a finite field of order $p^k$. 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 $k = 1$, the notions coincide.

Also, the multiplicative group of a finite field is always cyclic, even when the prime is $2$.

## Examples

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

Here, we discuss examples of prime powers.

### $n = 2^2 = 4$

In this case, the multiplicative group is $\{1,3 \}$, and is cyclic on $3$.

### $n = 3^2 = 9$

In this case, the multiplicative group is $\{ 1,2,4,5,7,8 \}$, and it has two possible generators: $2$ and $5$.