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.
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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.