Quasiprimitive Groups

Quasiprimitive

A permutation group is said to be quasiprimitive if all its non-trivial normal subgroups are transitive.

Classification

Let $G$ be a quasiprimitive permutation group on $\Omega$, and let $N=\mathrm{soc}(G)$, the socle of $G$ generated by all minimal normal subgroups. Then by ^9gl6q2, $G$ has at most two distinct transitive minimal normal subgroups, and by ^7p85qz we have $N=T^n$, where $T$ is simple and $n$ is a positive integer. Praeger shows that there are the following eight types.

• Holomorph type: ^mrwif1
  • (HS) holomorph simple: $N$ is a product of two minimal subgroups which are nonabelian simple
  • (HC) holomorph compound: $N$ is a product of two minimal normal subgroups which are not simple;
  • (HA) holomorph affine: $N$ is abelian
• Non-holomorph type:
  • (AS) almost simple: $N$ is simple, and $C_G(N)=1$;
  • (SD) simple diagonal: point stabilizer $N_{\omega }$ is simple and isomorphic to $T$;
  • (CD) compound diagonal: point stabilizer $N_{\omega }\cong T^k$ with $k⩾2$;
  • (TW) twisted wreath product: $N$ is nonabelian, non-simple, and regular;
  • (PA) product action: $N$ has no normal subgroup which is regular on $\Omega$.

Remark that HS, HC and HA are defined here, and they are primitive.

ref: Finite quasiprimitive graphs

PA type

$G$ preserves a product structure ${\Delta }^k$ on a $G$-invariant partition of $\Omega$ and the subgroup of $\mathrm{Sym}(\Delta )$ involved is quasiprimitive of type $AS$ with socle $T$. Thus a quasiprimitive group $G$ of type $PA$ induces a faithful product action on this partition of $\Omega$.

Define

$$N=T^k\le G\le H\wr S_k⩽\mathrm{Sym}(\Delta )\wr S_k$$

where $k>1$ and $T$ is a finite nonabelian simple group, $H$ is a quasiprimitive permutation group on $\Delta$ of type $AS$ with non-regular socle $T$, and $G$ acts transitively by conjugation on the simple direct factors of $N$.

Choose $\delta \in \Delta$ and set $R:=T_{\delta }$. Then $1<R<T$. There is a $G$-invariant partition ${\Omega }^{\prime }$ of $\Omega$ (which is possibly trivial in that the parts of ${\Omega }^{\prime }$ may have size 1) such that for some $\omega \in {\Omega }^{\prime },N_{\omega }=R^k$ and for $\alpha \in \omega ,N_{\alpha }$ is a subdirect product of $R^k$.

SD type

For $G$ to be primitive of type $AS$ the stabilizer $G_{\alpha }$ must be maximal in $G$, and in particular $G_{\alpha }\ne \left\{1_G\right\}$. A quasiprimitive permutation group of type $SD$ is a subgroup of the group

$$W=\{\left(a_1,\dots ,a_k\right)\pi :a_i\in \mathrm{Aut}(T),\pi \in S_k,a_i\equiv a_j(\mathrm{mod}\mathrm{Inn}(T))\text{ for all }i,j\},$$

where ${\pi }^{-1}\left(a_1,\dots ,a_k\right)\pi =\left(a_{1{\pi }^{-1}},\dots ,a_{k{\pi }^{-1}}\right)$ and $k>1$. The socle of $W$ is $\mathrm{soc}(W)=\left\{\left(t_1,\dots ,t_k\right):t_i\in \mathrm{Inn}(T)\right\}$, and a primitive action of $W$ on $T^{k-1}$ (which we identify with $\mathrm{Inn}(T{)}^{k-1}$ ) is defined by

$$\begin{array}{rl}\left(a_1,\dots ,a_k\right):\left(t_1,\dots ,t_{k-1}\right)& \mapsto \left(a_k^{-1}{t}_1{a}_1,\dots ,a_k^{-1}{t}_{k-1}{a}_{k-1}\right),\text{ and }\\ \pi :\left(t_1,\dots ,t_{k-1}\right)& \mapsto \left(t_{k{\pi }^{-1}}^{-1}{t}_{1{\pi }^{-1}},\dots ,t_{k{\pi }^{-1}}^{-1}{t}_{(k-1){\pi }^{-1}}\right),\end{array}$$

for $\left(a_1,\dots ,a_k\right)\pi \in W$ and $\left(t_1,\dots ,t_{k-1}\right)\in T^{k-1}$, where $t_k=1_T$. Thus for $\alpha =\left(1_T,\dots ,1_T\right)\in T^{k-1},W_{\alpha }=A\times S_k$ where $A=\{(a,\dots ,a):a\in \mathrm{Aut}(T)\}$.

$\Omega =T^{k-1}$, where $k>1$ and $T$ is a finite nonabelian simple group, $N=\mathrm{soc}(W)⩽G⩽W$ with the action defined above, and $G$ acts transitively by conjugation on the $k$ simple direct factors of $N$.

TW type

The twisted wreath product $T{\mathrm{twr}}_{\phi }P$ of groups $T$ and $P$ relative to $\phi$ may be defined as follows. Let $P$ have a transitive action on $\{1,\dots ,k\}$ and let $Q$ be the stabilizer of the point 1 in this action. Suppose that there is a homomorphism $\phi :Q\to \mathrm{Aut}(T)$ such that ${\mathrm{core}}_P({\phi }^{-1}(\mathrm{Inn}(T))=\left\{1_P\right\}$. Define

$$B:=\left\{f:P\to T:f(pq)=f(p{)}^{\phi (q)}\text{ for all }p\in P,q\in Q\right\}$$

Then $B$ is a group under pointwise multiplication and $B\cong T^k$. Let $P$ act on $B$ by

$$f^p(x):=f(px)\text{ for all }p,x\in P.$$

We define $T{\mathrm{twr}}_{\phi }P$ to be the semidirect product of $B$ by $P$ relative to this conjugation action of $P$. Such a twisted wreath product $T{\mathrm{twr}}_{\phi }P$ has a transitive action on $B$ such that $B$ acts by right multiplication and for $f\in B$ and $p\in P,p:f\mapsto f^p$.

TW

$\Omega =T^k=B$, where $k>1$ and $T$ is a finite nonabelian simple group, $G$ is a twisted wreath product $T{\mathrm{twr}}_{\phi }P$ defined as above, and $G$ acts on $\Omega$ with the action defined above.

Equivalent Condition

Proposition

For a imprimitive group $G⩽\mathrm{Sym}(\Omega )$, $G$ is quasiprimitive iff $G_{(\mathcal{B})}=1$ for any non-trivial block system $\mathcal{B}$.

\begin{proof} If $G$ is quasiprimitive and $G_{(\mathcal{B})}\ne 1$, then $G_{(\mathcal{B})}⊲G$ is transitive on $\Omega$, which is impossible.

Conversely, if $G$ is not quasiprimitive, then there exists $N⊲G$ such that $N$ is intransitive on $\Omega$. Then $N$-orbits form a block system $\mathcal{C}$ and $G_{(\mathcal{C})}⩾N\ne 1$, contradiction. \end{proof}