Group theory and generalizations Group actions and permutation groups Theorem

O'Nan–Scott theorem

Builds on Primitive permutation group Normal subgroup Simple group Automorphism group Wreath product Stabiliser General linear group

Statement

Let $G$ be a finite primitive permutation group on a set $\Omega$ . Let $N$ be the socle of $G$ , the subgroup generated by all minimal normal subgroups of $G$ . Then $N \cong T^k$ for a simple group $T$ and an integer $k \ge 1$ , and $G$ is of one of the following five types.

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 $\F_p^d$ .
Almost simple type: $N = T$ is nonabelian, and $T \le G \le \Aut(T)$ .
Diagonal type: $T$ is nonabelian and $k \ge 2$ . The stabiliser 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 $\abs{\Omega} = \abs{T}^{k-1}$ .
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 $N = \operatorname{soc}(H)^\ell$ .
Twisted wreath type: $T$ is nonabelian and $N$ acts regularly on $\Omega$ ; then $\abs{\Omega} = \abs{T}^k$ .

Remarks

The proof studies the minimal normal subgroups of $G$ . Each one is transitive by primitivity, and it is a direct power $T^m$ of a simple group. There 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 $N$ .

Examples: the symmetric group $S_n$ on $n \ge 5$ points is of almost simple type with socle $A_n$ . The group of maps $x \mapsto ax + b$ with $a \ne 0$ on $\F_p$ is of affine type with socle the translations. For $m \ge 5$ , the wreath product $S_m \wr S_2$ acting on pairs $(i, j)$ with $1 \le i, j \le m$ is of product type.

Books group the types in different ways, and some versions list eight classes instead of five. The theorem reduces many questions on finite primitive groups to questions on finite simple groups and their automorphism groups.

Going deeper

For a student

The socle of a primitive group is a power $T^k$ of one simple group, and the five types describe how $G$ acts around it. The affine type is linear algebra over $\F_p$ . The almost simple type is a simple group together with some of its automorphisms. The other three types are built from several copies of a nonabelian simple group. Combined with the classification of finite simple groups, the theorem is a main tool in two problems. One is to find the maximal subgroups of the symmetric and alternating groups, and the other is to bound the orders of primitive groups.

Reading path

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Tag	`Z4WVANZPK4`
URI	`m2://Z4WVANZPK4`
Kind	statement, theorem
Writing	W0 stub
MSC	20B15
Status	This copy of the site has no build data. Run `m2 status Z4WVANZPK4`.
About	Primitive permutation group
Uses	Primitive permutation group, Normal subgroup, Simple group, Automorphism group, Wreath product, Stabiliser, General linear group
Same as	Wikidata Q17099479 Wikipedia O'Nan–Scott theorem
Changed	2 November 2026