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 $A\space \mathrm{twr}_H \space G:= {_H B}{:}G$. $_HB$ is called the base group of the twisted wreath product. [!proposition] The base group $_HB$ of $A\space \mathrm{twr}_H \space G$ is isomorphic to $\Pi_{i\in I}A_i$ where $A_i=A$ and $|I|=|[G:H]|$. ^fdd6ea **Rmk.** The key point is, every $b\in {_HB}$ is determined by $g_ib$ 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_xb=(\overline xb)^{h_x}.

Since $H\curvearrowright A$ is a fixed group action, $xb$ is determined by $\overline xb$. By [[#^fdd6ea|^fdd6ea]], $A\space \mathrm{twr}_H \space G$ 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 $\mathrm{Inn}(T)\leq G\leq \mathrm{Aut}(T)$ for some nonabelian simple group $T$. ## Diagonal type [!definition] A group $G$ is said to be of diagonal type if $M\leq G\leq W$ and $G$ is primitive. Let $T$ be a nonabelian simple group and $k\geq 2$ 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 $M:=(\mathrm{Inn}(T))^k\leq G\leq W$. 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 $G$. ^2uq6rg ## Product type > [!definition] > > A group $G$ is said to be of product type if $M\leq G\leq W$ 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 $M:=K^n$.