Abstract Algebra and cyclic subgroups

[brydustin]

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

 

[fzero]

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 $$g^i$$ for some element $$g$$ that you haven't identified).