# 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."
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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."
user
The text passage is a discussion thread from a homework help forum, where a student is seeking assistance with proving that the group U(n^2 - 1) is not cyclic for integers n > 2. The student asks a specific question about finding the order of (n^2 - 1) and whether this is necessary for the proof. Another user provides guidance on how to approach the problem, suggesting the use of properties of the Euler's totient function and the Chinese Remainder Theorem.

TYPE: NO_QUESTION

However, upon closer inspection, the student does ask a specific mathematical question: "How does one find the order of (n^2 -1)? Is this information even needed to solve this problem?"

TYPE: SINGLE_QUESTION