= Sylow basis
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{title2=$\{P_p:p\mid|G|\},\quad P_pP_q=P_qP_p$}
= Sylow bases
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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 $P_pP_q=P_qP_p$, 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.
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