Template:Random hard fact

Thompson's critical subgroup theorem (FACT): Every group of prime power order has a critical subgroup: a characteristic subgroup that is commutator-in-center, Frattini-in-center, and self-centralizing.@@@Thompson's replacement theorem (FACT): If $$A$$ is an abelian subgroup of maximum order in a $$p$$-group $$P$$ and $$B$$ is an abelian subgroup such that $$A$$ normalizes $$B$$ but $$B$$ does not normalize $$A$$, we can replace $$A$$ be another abelian subgroup $$A^*$$ of maximum order that normalizes $$A$$ with $$A \cap B$$ a proper subgroup of $$A^* \cap B$$.@@@Sylow's theorem with operators (FACT): An analogue of Sylow's theorem where, instead of looking at $$p$$-subgroups, we consider the $$p$$-subgroups invariant under the action of a coprime automorphism group.