Hall subgroup
= Hall subgroup
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A Hall subgroup has order coprime to its index. For a set of primes $\pi$, a Hall $\pi$-subgroup has order using only primes of $\pi$ and index using only primes outside $\pi$. Thus its order contains the full prime-power contributions belonging to $\pi$. A <Sylow subgroup> is the single-prime instance. <Hall subgroup existence in soluble groups> generalizes Sylow existence to every prime set.