can someone please help? i'm studying for an exam and i'm stuck on this problem.
Prove that U(2^n) (n>less than 3) is not cyclic.
thanks so much
1)Prove that $\displaystyle 5^{2^{n-3}} \equiv 1 + 2^{n-1} ~ (2^n)$ for $\displaystyle n\geq 3$ by induction.
2)Now show that $\displaystyle 5^{2^{n-2}} \equiv 1 ~ (2^n)$ and $\displaystyle 5^{2^{n-3}} \not \equiv 1 ~ (2^n)$.
3)Show that $\displaystyle \{ \pm 5^k | 0\leq k < 2^{n-1} \} = U(2^n)$.
4)Therefore no element has order $\displaystyle \phi (2^n)$ in $\displaystyle U(2^n)$.