Socles and the Onan Scott Landscape — Examples

The translation subgroup $V={\mathbb{F}}_p\times \{1\}$ is elementary abelian of order $p$ and acts regularly by $x\mapsto x+b$.

Let $N⊴\mathrm{AGL}(1,p)$ be nontrivial. If $N\cap V\mathrm{\ne }1$, then $V\le N$ because $V$ has prime order. If $N\cap V=1$, then the image of $N$ in the quotient $\mathrm{AGL}(1,p)\mathrm{/}V\cong {\mathbb{F}}_p^{\times }$ is nontrivial; choose $(c,a)\in N$ with $a\mathrm{\ne }1$. For any $b\in {\mathbb{F}}_p$, the commutator $(b,1)(c,a)(-b,1)(c,a{)}^{-1}=((1-a)b,1)$ lies in $N\cap V$, contradiction. So every nontrivial normal subgroup contains $V$, and $V$ is the unique minimal normal subgroup.

Therefore [L1] applies, and the natural action of $\mathrm{AGL}(1,p)$ is of affine type.

The acting group is $A_n$ itself, and [A1] makes it a nonabelian simple group. Thus $A_n\le A_n\le \mathrm{Aut}(A_n)$, so [L1] shows that $A_n$ is almost simple.

In the natural primitive action, the socle is therefore $A_n$ itself, and this is the almost-simple branch of the O'Nan-Scott dictionary.

The socle of the acting group is $T\times T$, a direct product of two isomorphic nonabelian simple factors, and the point stabilizer is the diagonal subgroup $\mathrm{\Delta }(T)$ by construction. This stabilizer is maximal: if $\mathrm{\Delta }(T)<L\le T\times T$, choose $(a,b)\in L\setminus \mathrm{\Delta }(T)$. Multiplying by $(a^{-1},a^{-1})$ gives $(1,b{a}^{-1})\in L$ with $b{a}^{-1}\mathrm{\ne }1$. Conjugation by $\mathrm{\Delta }(T)$ and simplicity of $T$ then give $1\times T\le L$, and $\mathrm{\Delta }(T)(1\times T)=T\times T$. Thus $L=T\times T$.

A coset action is primitive exactly when its stabilizer is maximal, so step 1.1 makes this action primitive. Its socle and stabilizer are then exactly the defining features in [L1], and the action is a simple diagonal action.

The action of $S_5$ on five points is primitive and not regular, and the action of $S_2$ on the two coordinates is transitive. Therefore [L1] applies to $S_5\wr S_2$ on $\{1,2,3,4,5{\}}^2$.

The socle of the wreath product is $A_5^2$, which is nonabelian and acts coordinatewise. Thus the action is primitive and has the product-action socle data in [L2], rather than affine socle data.

The diagonal-type witness has two distinct minimal normal subgroups by [A1].

Applying [L1] to those two subgroups shows that both are regular. This is exactly the exceptional case allowed by the socle analysis.

Let $N$ be a minimal normal subgroup of $G$. The primitive-solvable fact [A1] shows that $N$ is elementary abelian and regular.

The uniqueness statement [A2] says that this minimal normal subgroup is the only one.

The socle is generated by the minimal normal subgroups, so with uniqueness from step 2.1 it equals $N$. Therefore the socle of $G$ is the unique regular elementary abelian minimal normal subgroup.

By [A1], the action is transitive and imprimitive.

By [A2], the subgroup $N=T\times T$ is minimal normal, but because $N$ preserves each copy of $T$ setwise, it has two orbits and is therefore not transitive.

Hence a transitive action can have a nontransitive minimal normal subgroup once primitivity is dropped.