Abstract Algebra and cyclic subgroups

In summary, the problem asks to prove that the nth roots of unity, defined as complex numbers z such that z^n = 1, form a cyclic subgroup of the complex numbers under multiplication of order n. This subgroup is closed under multiplication, associative, has an identity, and each element has an inverse. The definition of cyclic means that every element can be written as g^i for some element g. The purpose of this problem is to demonstrate the properties and characteristics of cyclic subgroups to readers.
  • #1
brydustin
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0

Homework Statement


from Algebra by Michael Artin, chapter 2, question 5 under section 2(subgroups)

An nth root of unity is a complex number z such that z^n =1. Prove that the nth roots of unity form a cyclic subgroup of C^(x) (the complex numbers under multiplication) of order n.


Homework Equations



cyclic subgroup = {z,z^2, ... , z^(n-1), z^n = 1}

Closed under multiplication, is associative, has an identity and is cyclic (z^(n+b) = z^b becausse z^n = 1).

The Attempt at a Solution



What's the point of this problem? Have I "proven" it, I'm trying to prepare for GRE and I am reviewing algebra (I'm a graduate, this isn't actually homework), but I don't see how its not immediately obvious that its a cyclic subgroup (put another way: What is Artin trying to show his readers?)
Thanks
 
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  • #2
You left out the requirement that every element has an inverse. You also have the definition of cyclic wrong (it's that every element can be written as [tex]g^i[/tex] for some element [tex]g[/tex] that you haven't identified).
 

FAQ: Abstract Algebra and cyclic subgroups

1. What is Abstract Algebra?

Abstract Algebra is a branch of mathematics that deals with the study of algebraic structures and their operations, such as groups, rings, and fields. It focuses on abstract and general concepts rather than specific numbers or equations.

2. What is a cyclic subgroup?

A cyclic subgroup is a subset of a group that can be generated by a single element through repeated application of the group's operation. This element is called a generator, and the subgroup contains all the powers of the generator and their inverses.

3. How do you determine if a subgroup is cyclic?

A subgroup is cyclic if it can be generated by a single element. This means that all the elements in the subgroup can be obtained by repeatedly applying the group's operation to the generator. If there is no such element, then the subgroup is not cyclic.

4. What is the order of a cyclic subgroup?

The order of a cyclic subgroup is the number of elements in the subgroup. It is equal to the number of powers of the generator that are required to generate all the elements in the subgroup.

5. How are cyclic subgroups used in Abstract Algebra?

Cyclic subgroups are used to study the structure and properties of larger groups. They can help determine if a group is abelian (commutative) or not, and they can also be used to classify groups into different categories. In addition, cyclic subgroups have applications in cryptography and other areas of mathematics.

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