Show $G$ is not simple – Sylow counts, element counts, and actions on cosets and on Sylow subgroups that produce a proper nontrivial normal subgroup.
Orbit-stabilizer – Lagrange, Cauchy, the orbit-stabilizer theorem, and the actions of \(G\) on itself, on its subgroups, and on cosets.
The class equation – the class equation, centers of \(p\)-groups, and Burnside’s lemma.
Sylow theory – existence, conjugacy, and the congruence and divisibility conditions on the number of Sylow subgroups.