91Ó°ÊÓ

. If \(H\) and \(K\) are finite groups whose orders are relatively prime, then Aut \((H \times K)\) \(\cong \operatorname{Aut}(H) \times\) Aut \((K)\). Show that this may fail if \((|H|,|K|)>1\). (Hint. Take \(H=\) \(\left.\mathbb{Z}_{p}=K .\right)\)

Prove that every group of order \(p^{2} q\), where \(p>q\) are primes, has a normal Sylow \(p\) -subgroup, and classify all such groups.