Tour:Multiplicative group modulo a prime is cyclic
This article adapts material from the main article: multiplicative group of a prime field is cyclic
This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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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
.
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