text: # Homework Help: prove that the group U(n^2 -1) is not cyclic
  
  1. Feb 29, 2012
  
  ### 2e4L
  
  Sorry if I formatted this thread incorrectly as its my first post ^^
  
  1. The problem statement, all variables and given/known data
  
  For every integer n greater than 2, prove that the group U(n^2 - 1) is not cyclic.
  
  2. Relevant equations
  
  3. The attempt at a solution
  
  I've done a problem proving that U(2^n) is not cyclic when n >3, but I'm failing to make a parallel.
  
  How does one find the order of (n^2 -1)? Is this information even needed to solve this problem?
  
  2. Feb 29, 2012
  
  ### Deveno
  
  you might want to prove some easier things first:
  
  suppose 8|k, that is:
  
  k = 2tm, where t ≥ 3.
  
  then φ(k) = φ(2t)φ(m), so
  
  U(k) ≅ U(2t) x U(m).
  
  use this to show U(k) cannot be cyclic, as it has a non-cyclic subgroup.
  
  now, consider even n and odd n for n2-1 separately.
  
  3. Feb 29, 2012
  
  ### 2e4L
  
  I have learned that property of Euler's totient function, but not the one about U(ab) = U(a) x U(b). I've come across it online, but have yet to use it in class.
  
  Is there another method?
  
  4. Feb 29, 2012
  
  ### Deveno
  
  U(ab) isn't always isomorphic to U(a) x U(b).
  
  for example U(4) is cyclic of order 2, but U(2) x U(2) has but a single element: (1,1).
  
  it IS true, if a and b are co-prime.
  
  this is actually a consequence of the chinese remainder theorem:
  
  if gcd(m,n) = 1, then [a]mn→([a]m,[a]n) is a group isomorphism of (Zmn,+) with (Zm,+) x (Zn,+).
  
  it's easy to check that [a]mn→([a]m,[a]n) is a ring homomorphism as well, hence Zmn and Zm x Zn have isomorphic groups of units (when...gcd(m,n) = 1. this is key).
  
  i find it odd, that you wouldn't use this result, since it builds on the result of your previous problem.
  
  5. Feb 29, 2012
  
  ### 2e4L
  
  It may be because it's still fairly early in the course. I learned the Chinese Remainder Theorem in a number theory class, but have yet to encounter it in my current one. We are going to study rings after the upcoming exam.
  
  The previous problem was in a different chapter of the text I believe. This question is supposed to be a "review."
explanation: The document is a forum post from a homework help thread where the user "2e4L" is seeking assistance with proving that the group \( U(n^2 - 1) \) is not cyclic for every integer \( n \) greater than 2. The post includes the problem statement, an attempt at a solution, and a discussion with another user "Deveno" who provides guidance and additional insights. The document contains a question and an ongoing discussion that serves as a partial answer.
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distilabel_metadata.content: # Homework Help: prove that the group U(n^2 -1) is not cyclic
  
  1. Feb 29, 2012
  
  ### 2e4L
  
  Sorry if I formatted this thread incorrectly as its my first post ^^
  
  1. The problem statement, all variables and given/known data
  
  For every integer n greater than 2, prove that the group U(n^2 - 1) is not cyclic.
  
  2. Relevant equations
  
  3. The attempt at a solution
  
  I've done a problem proving that U(2^n) is not cyclic when n >3, but I'm failing to make a parallel.
  
  How does one find the order of (n^2 -1)? Is this information even needed to solve this problem?
  
  2. Feb 29, 2012
  
  ### Deveno
  
  you might want to prove some easier things first:
  
  suppose 8|k, that is:
  
  k = 2tm, where t ≥ 3.
  
  then φ(k) = φ(2t)φ(m), so
  
  U(k) ≅ U(2t) x U(m).
  
  use this to show U(k) cannot be cyclic, as it has a non-cyclic subgroup.
  
  now, consider even n and odd n for n2-1 separately.
  
  3. Feb 29, 2012
  
  ### 2e4L
  
  I have learned that property of Euler's totient function, but not the one about U(ab) = U(a) x U(b). I've come across it online, but have yet to use it in class.
  
  Is there another method?
  
  4. Feb 29, 2012
  
  ### Deveno
  
  U(ab) isn't always isomorphic to U(a) x U(b).
  
  for example U(4) is cyclic of order 2, but U(2) x U(2) has but a single element: (1,1).
  
  it IS true, if a and b are co-prime.
  
  this is actually a consequence of the chinese remainder theorem:
  
  if gcd(m,n) = 1, then [a]mn→([a]m,[a]n) is a group isomorphism of (Zmn,+) with (Zm,+) x (Zn,+).
  
  it's easy to check that [a]mn→([a]m,[a]n) is a ring homomorphism as well, hence Zmn and Zm x Zn have isomorphic groups of units (when...gcd(m,n) = 1. this is key).
  
  i find it odd, that you wouldn't use this result, since it builds on the result of your previous problem.
  
  5. Feb 29, 2012
  
  ### 2e4L
  
  It may be because it's still fairly early in the course. I learned the Chinese Remainder Theorem in a number theory class, but have yet to encounter it in my current one. We are going to study rings after the upcoming exam.
  
  The previous problem was in a different chapter of the text I believe. This question is supposed to be a "review."
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a forum post from a homework help thread where the user "2e4L" is seeking assistance with proving that the group \( U(n^2 - 1) \) is not cyclic for every integer \( n \) greater than 2. The post includes the problem statement, an attempt at a solution, and a discussion with another user "Deveno" who provides guidance and additional insights. The document contains a question and an ongoing discussion that serves as a partial answer.
  
  Label: question-and-answer