Template:Random hard fact

<random>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.</random>