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  "body": "\n## Summary\n\nA Hall subgroup of a finite group is a [[subgroup|0X1JXT47GA]] whose order is [[coprime|0VZ48N083F]] to its [[index|0DTXBQPA2S]].\nHall subgroups generalise [[Sylow subgroups|0053CQE9Y2]] from one prime to a set of primes.\nIn a [[solvable group|0FXJ7YFMMX]] they always exist and behave like Sylow subgroups, by [[Hall's theorem|8Z8PE69SJS]]; in other groups they may fail to exist.\nThey are named after Philip Hall.\n\n## Explanations\n\n### L1 human\n\nA Sylow subgroup collects all the factors of one prime in the order of a group.\nA Hall subgroup does the same for a chosen set of primes.\nFor example, the group $A_5$ has order $60 = 2^2 \\cdot 3 \\cdot 5$.\nIts subgroup $A_4$ has order $12 = 2^2 \\cdot 3$ and index $5$, and $12$ and $5$ have no common factor, so $A_4$ is a Hall subgroup for the primes $2$ and $3$.\nBut $A_5$ has no subgroup of order $15$, so there is no Hall subgroup for the primes $3$ and $5$.\n",
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