[Math] A Nonabelian group of order of product of primes G has a trivial center – Fraleigh p. 153 15.18

Question

Using Exercise 37, show:
A nonabelian group G of order pq where p and q are primes has a trivial center.

Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hwi think 15solns.pdf

(1.) How do you envisage and envision to prove by contradiction? Why not direct proof?

(2.) What's the intuition?

Answer 1

Lemma: if $G/Z(G)$ is cyclic then $G$ is abelian.

So, only possible order for $G/Z(G)$ is $pq\implies$ $Z(G)=1$