# $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$

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.

• For a proof using a different approach, see math.stackexchange.com/questions/66043/…. – lhf Aug 28 '13 at 10:57