= Hall subgroup existence in soluble groups
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Every finite <soluble group> has a <Hall subgroup> for each prime set. Induct on the group order using an elementary abelian <minimal normal subgroup>. In the coprime hard case, lift a minimal normal subgroup of the quotient, choose its Sylow subgroup, and apply the <Frattini argument>. A proper normalizer reduces the order; a normal Sylow subgroup allows induction in its quotient. This proves existence without assuming an independent complement theorem.
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