Originally Posted by

**ziggychick** I should be able to do this one but I'm stuck...

I have a cyclic subgroup $\displaystyle G $ of order $\displaystyle q-1$ where $\displaystyle q$ is prime. i guess you meant "group"!

Let $\displaystyle p$ be a prime dividing $\displaystyle q-1$. Show G contains a unique cyclic subgroup of order $\displaystyle p$.

Now I know Cauchy's Theorem tells me G indeed has at least one subgroup of order $\displaystyle p$.

Also I know $\displaystyle G\simeq\mathbb{Z}_{pm}$ for some positive integer $\displaystyle m$.

I may be making this too complicated...Any hints?