Sylow Systems
A Sylow system is a set of Sylow p-subgroups Sp for each prime p such that SpSq = SqSp for all p and q. If we have a Sylow system, then the subgroup generated by the groups Sp for p in π is a Hall π-subgroup. A more precise version of Hall's theorem says that any solvable group has a Sylow system, and any two Sylow systems are conjugate.
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