# Abstract Algebra and cyclic subgroups

## 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

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).