# 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."