The O'Nan-Scott Theorem for Finite Primitive Permutation Groups, and Finite Representability

Let $G$ be a nontrivial finite primitive permutation group on $\Omega$. Then $G$ is permutation isomorphic to a group that is either of affine type, twisted wreath type, almost simple type, diagonal type, or product type.

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Affine type

Definition

A group $G$ is said to be of affine type if $V^{\ast }⩽G\le \mathrm{Aff}(k,p)$ and $G$ is primitive for some $k$-dimensional vector space $V$ over ${\mathbb{F}}_p$.

Let $V$ be a $k$-dimensional vector space over a field $\mathbb{F}$, i.e. $V={\mathbb{F}}_p^k$. Let $v\in V$. Define $v^{\ast }\in S^V$ by $x\mapsto x+v$. $v^{\ast }$ is called a translation. We define the affine group of $V$ to be $\mathrm{Aff}(k,p)\simeq {\mathbb{Z}}_p^k:\mathrm{GL}(k,p)$.

Note that $T^{-1}{v}^{\ast }T=(vT{)}^{\ast }$ for any $T\in \mathrm{GL}(V)$.

Twisted wreath type

Let $P$ be a transitive permutation group on $\{1,\cdots ,k\}$ where $k\ge 2$ and $Q$ be the stabilizer of $1$ in $P$. Suppose $\phi :Q\to \mathrm{Aut}(T)$ is a group homomorphism and $T$ is a simple nonabelian group with $\mathrm{Inn}(T)\le \phi (Q)$. Then $\phi$ is a group action of $Q$ on $T$. Define $G:=T{\mathrm{twr}}_{Q}P={}_{Q}B:P$ where ${}_{Q}B$ denotes the base group of the twisted wreath product.

Let $\Omega :=[G:P]$. $G$ is said to be of twisted wreath type if $G$ acts primitively on $\Omega$.

Since $G=T^{|[P:Q]|}:P$, then $G$ does NOT have $3$ $(P,P)$-double cosets. Hence it is not a group of rank $3$.

Wreath product

Let $G,A$ be groups. Let $\Omega$ be a set and $G$ act on it. Define $B:=A^{\Omega }=\{b:\Omega \to A\}.$Then we can define a group action $G\curvearrowright B$ by $\alpha b^g:={\alpha }^{g^{-1}}b$ for all $g\in G$, $\alpha \in \Omega$ and $b\in B$. Define $A{\wr }_{\Omega }G=B:G=A^{\Omega }:G$. $B$ is called the base group of the wreath product.

If $\Omega$ is a finite set, then $A{\wr }_{\Omega }G\simeq A^{|\Omega |}:G$.

Twisted Wreath Product

Let $G,A$ be groups. Let $H$ be a subgroup of $G$ acting on $A$. Take $\Omega =G$. Define ${}_{H}B:=\{b:G\to A:(xh)b=(xb{)}^h,\forall x\in G,h\in H\}.$It is easy to verify ${}_{H}B$ is a subgroup of $B$. Define $A{\mathrm{twr}}_{H}G:={}_{H}B:G$. ${}_{H}B$ is called the base group of the twisted wreath product.

Proposition

The base group ${}_{H}B$ of $A{\mathrm{twr}}_{H}G$ is isomorphic to ${\Pi }_{i\in I}{A}_i$ where $A_i=A$ and $|I|=|[G:H]|$.

Rmk. The key point is, every $b\in {}_{H}B$ is determined by $g_{i}b$ where $G={\cup }_{i\in I}{g}_{i}H$. For any $x\in G$, $x$ can be written as $\overline{x}{h}_x$ for some $h_x\in H$, $\overline{x}\in \{g_i:i\in I\}$. Then $xb=\overline{x}{h}_{x}b=(\overline{x}b{)}^{h_x}.$Since $H\curvearrowright A$ is a fixed group action, $xb$ is determined by $\overline{x}b$.

By ^fdd6ea, $A{\mathrm{twr}}_{H}G$ can be written as $A^{|[G:H]|}:G$, and it is just an action induced by $G\curvearrowright [G:H]$.

NOTE

The primitive groups of twisted wreath product type are much harder to find! The smallest such group is a permutation representation of $A_5\wr A_6$ on the cosets of a subgroup $H$ isomorphic to $A_5$, and has degree $60^6$.

Almost simple type

Definition

A group $G$ is said to be of almost simple type if $G$ is a finite almost simple primitive permutation group.

A finite group is almost simple if it is isomorphic to a group $G$ for which $\mathrm{Inn}(T)\le G\le \mathrm{Aut}(T)$ for some nonabelian simple group $T$.

Diagonal type

Definition

A group $G$ is said to be of diagonal type if $M\le G\le W$ and $G$ is primitive.

Let $T$ be a nonabelian simple group and $k\ge 2$ an integer. Let

$$A:=\{(a_1,\cdots ,a_k)\in (\mathrm{Aut}(T){)}^k:\mathrm{Inn}(T)a_i=\mathrm{Inn}(T)a_j,\forall i,j\}.$$

Then $A$ is a subgroup of $(\mathrm{Aut}(T){)}^k$. Let $W:=A:S_k$. Let $M:=(\mathrm{Inn}(T){)}^k\le G\le W$.

Suppose $H$ has socle $K$. Let $M:=K^n$. And the permutation is $G$ acting on $[G:D]$ where $D=\{(a,\cdots ,a)\pi \}\cong \mathrm{Aut}(T)\times S_k$. Note that $W$ has more than three $(D,D)$-double cosets, it is not rank $3$ and so for $G$.

Product type

Definition

A group $G$ is said to be of product type if $M\le G\le W$ and $G$ is primitive. When $H$ is of almost simple type or diagonal type, $G$ said to be of almost simple product type or diagonal product type respectively.

Let $H$ be a primitive permutation group on $\Gamma$. Let $\Delta :=\{1,\cdots ,n\}$. Define $W:=H{\wr }_{\Delta }{S}_n\simeq H^n:S_n$. Then $W$ acts on $\Omega ={\Gamma }^n$. Suppose that $H$ has socle $K$. Let $M:=K^n$.