This question has been answered a lot of times on this site, but I'm looking for an approach that does not use Sylow theory, since this is not covered in my syllabus. All answers I read this far used material that I did not yet learn. My level this far is up to automorphisms, group actions, and the isomorphism theorems.
My syllabus uses the following construction of a non-abelian group of order pqpq where q|p−1q|p−1. Let N=CpN=Cp such that Aut(N)Aut(N) has order p−1p−1. From Cauchy's theorem we deduce that there exists a subgroup H⊂Aut(N)H⊂Aut(N) of order qq. Let τ:H→Aut(N)τ:H→Aut(N) be the identity map. Then N⋊τHN⋊τH has order pqpq and is non-abelian.
This far I can follow, but now I have to show that this group is the only non-abelian group of order pqpq. A hint for this exercise is to use that (Z/pZ)∗(Z/pZ)∗ is cyclic if pp is prime.
My attempt (it is not really an attempt, I just looked what I could deduce, but it led me nowhere): Let