Imprimitive Rank 3 Group is a Wreath Product of two 2-Transitive Groups

Transclude of On-imprimitive-rank-3-permutation-groups#^him0rv

Lemma

Let $B$ be a set and suppose that $G⩽\mathrm{Sym}(B)\wr S_n$ is transitive of rank $3$. Then both $G_B^B$ and the projection $X$ of $G$ to $S_n$ are $2$-transitive.

\begin{proof} Let $\alpha \in B$. Then $G_{(\alpha ,1)}$ has $3$ orbits on the points of $B\times \{1,\cdots ,n\}$. Since $B_1=B\times \{1\}$ is a block for $G$, any orbit intersecting non-trivially with $B_1\setminus \{(\alpha ,1)\}$ is contained in $B_1$. Hence, the $3$ orbits of $G_{(\alpha ,1)}$ are $\{(\alpha ,1)\}$, $B_1\setminus \{(\alpha ,1)\}$ and $B\times \{2,\cdots ,n\}$. It yields that $(G_B^B{)}_{\alpha }$ is transitive on $B\setminus \{\alpha \}$ and so $G_B^B$ is $2$-transitive on $B$. Moreover, as $(G^{\mathcal{B}}{)}_1$ is transitive on $\{2,\cdots ,n\}$, we have $G^{\mathcal{B}}$ is $2$-transitive. \end{proof}

ref: On imprimitive rank 3 permutation groups