• Support PF! Buy your school textbooks, materials and every day products Here!

Abstract Algebra and cyclic subgroups

  • Thread starter brydustin
  • Start date
  • #1
205
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
 

Answers and Replies

  • #2
fzero
Science Advisor
Homework Helper
Gold Member
3,119
289
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).
 

Related Threads on Abstract Algebra and cyclic subgroups

  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
1
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
8
Views
3K
Replies
1
Views
1K
Replies
4
Views
2K
Replies
3
Views
9K
Replies
2
Views
776
Top