O'Nan-Scott Theorem

O'Nan, Scott

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.

ref: The O’Nan-Scott Theorem for Finite Primitive Permutation Groups, and Finite Representability

Sketch of the proof:

• take $N=\mathrm{soc}(G)$, and recall that $N$ is a direct product of two minimal normal subgroups or $G$ has exactly one minimal normal subgroup.
• if $N$ is regular (holomorph)
  • $N$ is abelian: affine type
  • $N$ is nonabelian: twisted wreath type
• if $N$ is not regular, then $N_{\alpha }$ is non-trivial (non-holomorph)
  • $N$ is simple: almost simple type
  • $N\cong T^k$ for some nonabelian simple $T$ and $N$ is primitive: diagonal type
  • $N\cong T^k$ and $N$ is imprimitive: product type