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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], let $p$ be a [[prime number|01826KMG2H]], and write $\\abs{G} = p^n m$ with $n \\ge 0$ and $p \\nmid m$.\nA [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$ is a subgroup of order $p^n$, and $n_p$ denotes the number of them.\n\n1. $G$ has a Sylow $p$-subgroup.\n2. Any two Sylow $p$-subgroups of $G$ are [[conjugate|0H11ZPX4ND]] in $G$, and every [[$p$-subgroup|06QYRTX9KQ]] of $G$ is contained in some Sylow $p$-subgroup.\n3. $n_p \\equiv 1 \\pmod p$, $n_p$ divides $m$, and $n_p = [G : N_G(P)]$ for any Sylow $p$-subgroup $P$, where $N_G(P)$ is the [[normaliser|0YGN3VAA0D]] of $P$.\n\n## Remarks\n\nEach part has its own node: the [[first Sylow theorem|0V98WY94B8]], the [[second Sylow theorem|09GCQ5STVJ]] and the [[third Sylow theorem|0Q8DGAZSJ4]].\nThe rest of part 3 is in the nodes [[$n_p$ divides the index|0M3PYJ3TSE]] and [[$n_p$ is the index of the normaliser|09ZJ6CAK7S]].\nThe usual proofs let a group act on a set and count orbits or fixed points.\nPart 1 uses the action on subsets of size $p^n$, or induction on $\\abs{G}$.\nParts 2 and 3 let a $p$-group act on conjugates or cosets, and count its fixed points modulo $p$.\n\nExample: every group $G$ of order $15 = 3 \\cdot 5$ is cyclic.\nBy part 3, $n_3$ divides $5$ and $n_3 \\equiv 1 \\pmod 3$, so $n_3 = 1$; and $n_5$ divides $3$ and $n_5 \\equiv 1 \\pmod 5$, so $n_5 = 1$.\nSo $G$ has a [[normal|0CB0DX5BVV]] subgroup of order $3$ and one of order $5$.\nTheir intersection is $\\set{e}$, since its order divides both $3$ and $5$.\nHence $G \\cong \\Z/3\\Z \\times \\Z/5\\Z \\cong \\Z/15\\Z$.\n\nThe prime power in part 1 matters: the converse of [[Lagrange's theorem|3C53TD]] fails for other divisors, since [[$A_4$ has no subgroup of order $6$|0PDBVPP233]].\n\n## Explanations\n\n### L1 human\n\nLagrange's theorem says that the order of a subgroup divides the order of the group, but a group need not have a subgroup of each such order.\nThe Sylow theorems say that this does hold for the largest prime power dividing the order.\nMoreover, these largest prime-power subgroups are all alike, and their number obeys strict rules.\n\n### L2 human\n\nThe Sylow theorems are the first main tool to study finite groups through their order alone.\nA typical use: the counting rules in part 3 force $n_p = 1$, so the Sylow subgroup is normal and the group is not simple.\nIn this way one shows, for example, that no group of order $20$ and no group of order $pq$, with primes $p \u003c q$, is simple.\nThe theorems lead to the [[Frattini argument|0QTZDKJEDV]], to [[Hall subgroups|KPWW9X14V2]] in solvable groups, and to the study of how $p$-elements are conjugate, as in the [[focal subgroup theorem|H1BSG0J5G1]].\n",
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