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

TL;DR

This paper investigates the embeddings of certain non-generic primitive simple groups into exceptional algebraic groups over fields of characteristic not dividing their order, analyzing their conjugacy classes and maximality properties.

Contribution

It provides explicit constructions and classifications of embeddings of specific simple groups into exceptional Lie type groups, highlighting their primitivity, conjugacy, and maximal subgroup status.

Findings

Constructed embeddings of PSL_2(q) in exceptional groups for specified q.

Determined the number of conjugacy classes of these subgroups.

Showed certain subgroups are strongly imprimitive and not maximal.

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 $\overline{G}$.