# Cyclic Groups

Introduction

Cyclic groups are one of the key building blocks for groups - especially abelian groups. The finite cyclic groups take several common forms including the integers mod n for positive integers n. This is denoted by Z/nZ, where Z is the ring of integers.

Definition

A finite group G is called a cyclic group if it is generated by a single element. The way to write this symbolically is G = <g>, where the notation <g> stands for all powers of g. (Don't forget, even though we'll sometimes use words like 'powers' which sounds like we're talking about multiplication, we are talking about a generic group.) Another way to think about this is that if G is cyclic, and g is a generator of order n, then all the elements in the group are {e, g, g2, g3, ..., gn-1}. The word cyclic is used because the sequence g, g^2, g^3, ... eventually cycles around and begins to repeat.

There are two types of cyclic groups: the integers Z under addition, and the integers mod n under addition.

Example

If you look at the solutions to the equation xn-1=0, you get what's called the "nth roots of unity." If you multiply one nth root of unity by another, you get a third nth root of unity. So the nth roots of unity form a group under multiplication. It is a cyclic group because the complex number e2πi/n generates the group.

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Course Description
Abstract Algebra deals with groups, rings, fields, and modules. These are abstract structures which appear in many different branches of mathematics, including geometry, number theory, topology, and more. They even appear in scientific topics such as quantum mechanics.