Show that $(\mathbb{Z}/2^n \mathbb{Z})^{\times}$ is not cyclic for any $n > 3.$

Question

I need some help with the following question.

Show that $(\mathbb{Z}/2^n \mathbb{Z})^{\times}$ is not cyclic for any $n > 3.$

There is the following hint in my book: find two distinct subgroups of order $2.$

But I don't see any subgroups of order $2.$ Is group $(\mathbb{Z}/2 \mathbb{Z})^{\times}$ subgroup of $(\mathbb{Z}/2^n \mathbb{Z})^{\times}?$ Show the second subgroup, please.

Answer 1

Since $(2^n-1)^2 \equiv 1 \pmod {2^n}$, $\langle 2^{n}-1 \rangle$ is a subgroup of order 2. Also, since $n \geq 3$, $(1+2^{n-1})^2 \equiv 1 \pmod{2^n}$, so $\langle 1+2^{n-1} \rangle$ has order 2. It should be easy to see that these subgroups are distinct, as $n \neq 1$.