Problem.
Let $G$ be a group of order $pq$ such that $p$ and $ q$ are prime integers.
I am to show that every proper subgroup of $G$ is cyclic.
My attempt.
What I know: Any element $a$ divides $pq$ and $a^{pq} = e$.
The order of subgroups $H$ divide $pq$ by Lagrange.
If I could show that $G$ is cyclic, then all subgroups must be cyclic.
If I can show that $G$ is a group of prime order, then I can show that it is cyclic.
I'm not sure what more I can do here…I've tried looking at Fermat's Little Theorem but I can't seem to properly understand it and how it could affect my problem..
Best Answer
Hint: Every proper nontrivial subgroup have prime orders.
If $n=p_1^{a_1}p_2^{a_2}\dots p_t^{a_t}$ ,where $p_i$'s are different primes,then the only divisors of $n$ are of the form $p_1^{b_1}p_2^{b_2}\dots p_t^{b_t}$ where $0\leq b_i \leq a_i$ for all $i=1,2,\dots,t$.