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  "body": "\n## Definition\n\nLet $G$ be a finite [[group|G5S1M9]].\nA *Hall subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $H$ whose [[order|0B85S2V2DR]] and [[index|0DTXBQPA2S]] are [[coprime|0VZ48N083F]]: $\\gcd(\\abs{H}, [G : H]) = 1$.\n\nLet $\\pi$ be a set of [[prime numbers|01826KMG2H]].\nA *$\\pi$-number* is a positive integer whose prime factors all lie in $\\pi$.\nA *Hall $\\pi$-subgroup* of $G$ is a subgroup $H$ such that $\\abs{H}$ is a $\\pi$-number and $[G : H]$ has no prime factor in $\\pi$.\n\n## Remarks\n\nA Hall subgroup $H$ is a Hall $\\pi$-subgroup for $\\pi$ the set of primes dividing $\\abs{H}$.\nFor $\\pi = \\set{p}$, the Hall $\\pi$-subgroups are the [[Sylow $p$-subgroups|0053CQE9Y2]].\n\nExample: $A_4 \\le A_5$.\nHere $\\abs{A_4} = 12$ and $[A_5 : A_4] = 5$, and $\\gcd(12, 5) = 1$, so $A_4$ is a Hall $\\set{2, 3}$-subgroup of $A_5$.\n\nNon-example: the subgroup $\\set{0, 2}$ of $\\Z/4\\Z$.\nIts order is $2$ and its index is $2$, and $\\gcd(2, 2) = 2 \\ne 1$.\n\nHall $\\pi$-subgroups need not exist: $A_5$ has no subgroup of order $15$ or $20$, so it has no Hall $\\set{3, 5}$-subgroup and no Hall $\\set{2, 5}$-subgroup.\nIn solvable groups they always exist, by [[Hall's theorem|8Z8PE69SJS]].\nA normal Hall subgroup always has a complement, by the [[Schur–Zassenhaus theorem|64K1A69C80]].\n",
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