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

  1. 1. The problem statement, all variables and given/known data
    For any integer n>2, show that there are at least two elements in U(n) that satisfy x^2 = 1.


    2. Relevant equations
    None


    3. 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.
     
  2. jcsd
  3. 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.
     
    Last edited: Feb 9, 2011
  4. 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 [itex] (\mathbb{Z}/n\mathbb{Z})^{\times} [/itex]. So how do you write -1 in [itex] \mathbb{Z}/n\mathbb{Z} [/itex]?
     
  5. Dick

    Dick 25,628
    Science Advisor
    Homework Helper

    (n-1)^2=n^2-2n+1. Reduce those three terms mod n.
     
  6. 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?
     
  7. Dick

    Dick 25,628
    Science Advisor
    Homework Helper

    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?
     
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