= Hall conjugacy and embedding in finite soluble groups
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= Hall theorem for soluble groups
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For any prime set $\pi$, every finite soluble group has Hall $\pi$-subgroups, any two are conjugate, and every $\pi$-subgroup lies in one. Induction uses an elementary abelian minimal normal subgroup $N$ of characteristic $p$. If $p\in\pi$, the preimage of a quotient Hall subgroup is itself a Hall subgroup. If $p\notin\pi$, <coprime splitting over an elementary abelian normal subgroup> gives a conjugate class of complements in that preimage. The same complement conjugacy in the preimage of a smaller quotient subgroup gives the embedding assertion.
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