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  "body": "\n## Statement\n\nLet $G$ be a finite [[primitive permutation group|Y10S0751KG]] on a set $\\Omega$.\nLet $N$ be the socle of $G$, the subgroup generated by all minimal [[normal subgroups|0CB0DX5BVV]] of $G$.\nThen $N \\cong T^k$ for a [[simple group|0YMYMNQ0QZ]] $T$ and an integer $k \\ge 1$, and $G$ is of one of the following five types.\n\n1. Affine type: $N$ is an elementary abelian group of order $p^d$ for a prime $p$, and $N$ acts regularly on $\\Omega$. Then $\\Omega$ can be identified with $\\F_p^d$ so that $G = N \\rtimes G_0$, where $N$ is the group of translations and $G_0 \\le \\mathrm{GL}_d(\\F_p)$ acts irreducibly on $\\F_p^d$.\n2. Almost simple type: $N = T$ is nonabelian, and $T \\le G \\le \\Aut(T)$.\n3. Diagonal type: $T$ is nonabelian and $k \\ge 2$. The [[stabiliser|0AYMP41THN]] in $N$ of a point is a diagonal subgroup $\\set{(\\alpha_1(t), \\dots, \\alpha_k(t)) : t \\in T}$, where the $\\alpha_i$ are automorphisms of $T$. Then $\\abs{\\Omega} = \\abs{T}^{k-1}$.\n4. Product type: there are an integer $\\ell \\ge 2$ and a primitive group $H$ on a set $\\Delta$, of almost simple or diagonal type, with these properties. One has $\\Omega = \\Delta^\\ell$ and $G \\le H \\wr S_\\ell$ in its product action on $\\Delta^\\ell$. The socle is $N = \\operatorname{soc}(H)^\\ell$.\n5. Twisted wreath type: $T$ is nonabelian and $N$ acts regularly on $\\Omega$; then $\\abs{\\Omega} = \\abs{T}^k$.\n\n## Remarks\n\nThe proof studies the minimal normal subgroups of $G$.\nEach one is transitive by primitivity, and it is a direct power $T^m$ of a simple group.\nThere are at most two minimal normal subgroups, and the type is read off from how a point stabiliser meets $N$ and how $G$ permutes the $k$ factors of $N$.\n\nExamples: the symmetric group $S_n$ on $n \\ge 5$ points is of almost simple type with socle $A_n$.\nThe group of maps $x \\mapsto ax + b$ with $a \\ne 0$ on $\\F_p$ is of affine type with socle the translations.\nFor $m \\ge 5$, the [[wreath product|W1X984PKJ9]] $S_m \\wr S_2$ acting on pairs $(i, j)$ with $1 \\le i, j \\le m$ is of product type.\n\nBooks group the types in different ways, and some versions list eight classes instead of five.\nThe theorem reduces many questions on finite primitive groups to questions on finite simple groups and their automorphism groups.\n\n## Explanations\n\n### L1 human\n\nPrimitive permutation groups are the indivisible pieces of all finite permutation groups.\nThe O'Nan–Scott theorem says that they come in only five shapes.\nEach shape is described by a simple group $T$ and the way copies of $T$ sit inside the group.\nAfter this theorem, studying primitive groups mostly means studying simple groups.\n\n### L2 human\n\nThe socle of a primitive group is a power $T^k$ of one simple group, and the five types describe how $G$ acts around it.\nThe affine type is linear algebra over $\\F_p$.\nThe almost simple type is a simple group together with some of its automorphisms.\nThe other three types are built from several copies of a nonabelian simple group.\nCombined with the classification of finite simple groups, the theorem is a main tool in two problems.\nOne is to find the maximal subgroups of the symmetric and alternating groups, and the other is to bound the orders of primitive groups.\n",
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