What are some difficult, challenging and fair exercises in group theory? I know it is quite general, in particular I am referring to these areas of group theory:
- theory of automorphism
- group actions
- Sylow's theorem
- semi-direct products
- simple groups
(an exercise of the type I mean can be: " if p and q are distinct primes, show that a group of order $p^3q$ is not simple" or also "What is $Aut(Q)$? Where $Q$ the group of quaternions")