Classification of Quasiprimitive Groups

Quasiprimitive

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

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 ^7p85qz, there is $N=T^n$, 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