A Hall subgroup has order coprime to its index. For a set of primes , a Hall -subgroup has order using only primes of and index using only primes outside . Thus its order contains the full prime-power contributions belonging to . A Sylow subgroup is the single-prime instance. Hall subgroup existence in soluble groups generalizes Sylow existence to every prime set.
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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A Hall subgroup is a concept from group theory, specifically in the study of finite groups. It is named after Philip Hall, who introduced the concept in his work on groups and combinatorics.