Affine Quasiprimitive is Primitive

Proposition

Let $G⩽\mathrm{AGL}(V)$ be an affine group. If $G$ is quasiprimitive on $V$, then $G$ is primitive.

\begin{proof} Suppose that $G=V:G_0⩽V:\mathrm{GL}(V)$. For any $v\in V$ and $g\in G_0⩽\mathrm{GL}(V)$, define $vg:w\mapsto (v+w{)}^g$.

If $G_0$ is irreducible, then $G$ is primitive. Therefore, $G$ imprimitive yields $G_0$ reducible. Then there is a subspace $W\subseteq V$ stabilized by $G_0$. However, $W⊲G$ is not transitive on $V$ and so $G$ is not quasiprimitive, contradiction. \end{proof}

Another proof comes from Group Theory - 2024spring.

\begin{proof} Suppose $G$ is imprimitive on $V$. Then by Equivalent Condition of Primitivity, $G_0$ is not a maximal subgroup and there is a $M$ satisfying $G_0<M<G$. Let $N:=M\cap V$. Note that $N$ is a normal subgroup of $V$ by $V$ abelian. Also we have $N=M\cap V⊲M$ by $V⊲G$. It yields that $N⊲⟨M,V⟩=G$. Then $N⊲G$ is not transitive on $V$, which contradicts to $G$ quasiprimitive. \end{proof}

Remark. In fact, the group $G$ is of type HA in the Classification of Quasiprimitive Groups.