TITLE: $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$
QUESTION [9 upvotes]: Question is to Prove that $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not  cyclic Group for $n\geq 3$.
Hint : Find two subgroups of order $2$.
I somehow feel that a cyclic group can not have two distinct groups of same order. but, I am not sure about the proof.
I have no idea how to proceed for this. 
any hint would be appreciated.
Thank You.

REPLY [2 votes]: Consider $\pm(2^{k-1}+1)$. What is the order of these two elements if $k\geqslant 3$? Note they are $\neq \pm 1$ if $k\geqslant 3$.