Classification of natural numbers for which the multiplicative group is cyclic

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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.

Related facts

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.