Abstract Algebra: Question About the Elements in U(n)

[Shoney45]

Homework Statement

For any integer n>2, show that there are at least two elements in U(n) that satisfy x^2 = 1.

Homework Equations

None

The Attempt at a Solution

If the definition of the group U(n) is "the set of all positive integers less than n and relatively prime to n" then the group U(5) has elements {1,2,3,4}. Clearly then, 1^2 = 1. But I can't drum up any way to get any other elements in the group to equal 1 when squared.

The set U(n) is a group under modulo n, but I am pretty weak on modular arithmetic, and am trying to spend the day boning up on it. So for now, I don't understand how to find another element such that x^2 = 1.

 

[Shoney45]

I'm starting to figure this one out. I chose the case where n = 5. So the set of elements for U(n) = {1,2,3,4}. And under mod 5, 1^2 - 1 and 4^2 = 1.

I'm trying other mod n's, and it is looking like the integers which will equal 1 when squared are 1, and the largest integer in the set.

I don't know how to generalize this for the arbitrary case though. But I'm trying to screw around with a quadratic equation to see where I can go. Since the largest element in U(n) = n-1, I am looking for an mod n such that (n-1)^2 = 1. But I'm not getting very far with this line of thought.

 

[spamiam]

Here's one way to look at it. In the integers, the equation x² = 1 has two solutions, namely 1 and -1. Well, this is actually true in U(n) as well, since U(n) is really the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^{\times}$. So how do you write -1 in $\mathbb{Z}/n\mathbb{Z}$?

 

[Dick]

(n-1)^2=n^2-2n+1. Reduce those three terms mod n.

 

[Shoney45]

(n-1)^2=n^2-2n+1. Reduce those three terms mod n.

I've been searching through my book and the internet, and I can't figure out how to do this. I never learned modular arithmetic in my proofs class, so I am having to learn it on the fly in my modern algebra class. Is there a step in the right direction you could give me on this one?

 

[Dick]

Two numbers are equal mod n if they differ by an even multiple of n. Can you tell if 0 and n^2 are equal mod n?