Embeddings of $\mathrm{PSL}_2(q)$ in exceptional groups of Lie type over a field of characteristic $\ne2,3$

Abstract

Let $\boldsymbol{G}$ be an algebraic group of exceptional Lie type in characteristic $p$, $G=\boldsymbol{G}^{\sigma}$ its fixed-point subgroup under the action of a Steinberg endomorphism $\sigma$, and $\overline{G}$ an almost simple group with socle $G$. A maximal subgroup $M<\overline{G}$ is called non-generic if it is almost simple and its socle is not isomorphic to a group of Lie type in characteristic $p$. A finite subgroup $H$ of $\boldsymbol{G}$ is Lie primitive if it does not lie in any proper closed positive-dimensional subgroup of $\boldsymbol{G}$; it is Lie imprimitive if it lies in a positive-dimensional subgroup $\boldsymbol{X}$ of $\boldsymbol{G}$; it is strongly imprimitive if $\boldsymbol{X}$ can be chosen to be stable under the action of $N_{\mathrm{Aut}\boldsymbol{G}}(H)$, where $\mathrm{Aut}\boldsymbol{G}$ is the group generated by inner, diagonal, graph, and field automorphisms of $\boldsymbol{G}$. We study the possible embeddings of a non-generic primitive simple group $H$ in the adjoint algebraic group $\boldsymbol{G}$ in characteristic coprime to $|H|$ when $(\boldsymbol{G},H)$ is one of $(\boldsymbol{F}_4,\mathrm{PSL}_2(25))$, $(\boldsymbol{F}_4,\mathrm{PSL}_2(27))$, $(\boldsymbol{E}_7,\mathrm{PSL}_2(29))$, $(\boldsymbol{E}_7,\mathrm{PSL}_2(37))$. In particular, we construct copies of $H$ in $G$ over a suitable finite field $k$, and use them to deduce the number of conjugacy classes of $H$ in $G$ and $\boldsymbol{G}$, and whether $N_{\overline{G}}(H)$ is a maximal subgroup of $\overline{G}$. We also study the case of $H\simeq\mathrm{Alt}_6\simeq\mathrm{PSL}_2(9)$ when $\boldsymbol{G}$ is one of $\boldsymbol{F}_4$ and $\boldsymbol{E}_6$ in characteristic coprime to $|H|$, and show that in such cases $H$ is a strongly imprimitive subgroup of $\boldsymbol{G}$; in particular, $N_{\overline{G}}(H)$ is not a maximal subgroup of $\overline{G}$.