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.

Statement

In the language of modular arithmetic

Let $p$ be a prime number. Then, the multiplicative group modulo $p$ is a cyclic group of order $p-1$. In other words, it is isomorphic to the group of integers modulo $p-1$.

A generator for this multiplicative group is termed a primitive root modulo $p$. 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 $2$ is the trivial group, so this is not an interesting case.

The prime 3

The multiplicative group modulo $3$ is of order two, and the element $2$ is a primitive root in this case.

The prime 5

The multiplicative group modulo $5$ is of order four. The element $2$ is a primitive root. The powers of $2$ include all elements: $2^{0}=1,2^{1}=2,2^{2}=4,2^{3}=3$.

$3$ is also a primitive root. Its powers are $3^{0}=1,3^{1}=3,3^{2}=4,3^{3}=2$.

The prime 7

The multiplicative group modulo $7$ is of order six. $2$ is not a primitive root: it has order $3$, and its powers only include $1,2,4$. On the other hand, $3$ is a primitive root, with $3^{0}=1,3^{1}=3,3^{2}=2,3^{3}=6,3^{4}=4,3^{5}=5$.

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