# Tour:Multiplicative group modulo a prime is cyclic

**This article adapts material from the main article:** multiplicative group of a prime field is cyclic

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WHAT YOU NEED TO DO:

- Understand the statement of the result.
- Understand how the statement is illustrated by each of the examples.

## Contents

## Statement

### In the language of modular arithmetic

Let be a prime number. Then, the multiplicative group modulo is a cyclic group of order . In other words, it is isomorphic to the group of integers modulo .

A generator for this multiplicative group is termed a primitive root modulo . While the theorem states that primitive roots exist, there is no procedure or formula known for obtaining a primitive root.

## Examples

### The prime 2

The multiplicative group modulo is the trivial group, so this is not an interesting case.

### The prime 3

The multiplicative group modulo is of order two, and the element is a primitive root in this case.

### The prime 5

The multiplicative group modulo is of order four. The element is a primitive root. The powers of include all elements: .

is also a primitive root. Its powers are .

### The prime 7

The multiplicative group modulo is of order six. is *not* a primitive root: it has order , and its powers only include . On the other hand, is a primitive root, with .

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Elements of multiplicative group equal generators of additive group|UP: Introduction four (beginners)|NEXT: Factsheet four (beginners)