The problem is:
For every integer $n$ greater than 2, prove that the group $U(n^2-1)$ is not cyclic.
I have proved this for $U(2^n)$ where $n>2$ by showing more than one element of order 2 exists, which is a contradiction to Eulers phi function, making it noncyclic. As I try to apply similar logic to this problem, I seem to get stuck. Any ideas on how to manipulate this?
$gcd(n,n^2-1) = 1 \quad $ since $n(n) + (-1)*(n^2-1) = 1$ so $n \in U(n)$
Recall a cyclic group can have at most 1 subgroup of order 2. Consider the orders of the elements $\pm1 $ and $\pm n$ What are the orders of $<-1>, <n>, <-n>$