Question

For every integer n greater than 2, Prove that the group U(n^2-1) is not cyclic.

Answer 1

Maybe proof by contrapositive.

Answer 2

I'm not sure what you mean?

Answer 3

Okay, first what is your definition of cyclic group?

Answer 4

generated by an element and the gcd between the generator and an element in the cyclic group has to be 1

Answer 5

So all elements are coprime so some generator in the set?

Answer 6

yes, at least I believe that's right

Answer 7

Hmmm, well we know all the numbers in the group can be factored as \((n-1)(n+1)\)

Answer 8

ok

Answer 9

so how would I answer this?

Answer 10

\(U(n)\) if i remember correctly is the group of units of integers modulo \(n\)
is that correct?

Answer 11

in that case there are 4 elements \(g\) with \(g^2=1\) i.e. 4 elements that are there own inverses

Answer 12

which i guess i should add is not possible in a cyclic group.
since \(n^2-1=(n+1)(n-1)\) we have as units \(1,-1,n,-n\)

Answer 13

So my answer is that since a cyclic group can't have the elements as their own inverses, then this group cannot be cyclic?