 Sylow basis

A Sylow basis consists of one Sylow subgroup for each prime dividing the group order, with any two of them permutable as subgroups. Permutable means $P_p{P}_q=P_q{P}_p$, not elementwise commutation. Their product over any subset of primes is a Hall subgroup. Every finite soluble group has such a basis, and all bases are simultaneously conjugate. Induction over an elementary abelian minimal normal subgroup proves these facts using coprime splitting over an elementary abelian normal subgroup: lift the other-prime quotient basis into a complement, and lift the characteristic-prime Sylow subgroup by taking its full preimage.