论文标题
$ \ mathrm {psl} _2（q）的嵌入在特征$ \ ne2,3 $的领域的特殊类型中
Embeddings of $\mathrm{PSL}_2(q)$ in exceptional groups of Lie type over a field of characteristic $\ne2,3$
论文作者
论文摘要
令$ \ boldsymbol {g} $为特征性$ p $，$ g = \ boldsymbol {g}^σ$的固定点子组在Steinberg endopoint子组中的一个代数类型，在Steinberg Endomorphism $σ$和$ fline etlline {g} $ a socle $ g $的情况下。最大亚组$ m <\ overline {g} $几乎很简单，并且其socle对特征性$ p $中的谎言类型不是同构。 $ \ boldsymbol {g} $的有限亚组$ h $如果不在$ \ boldsymbol {g} $的任何适当的闭合正维子群中，则是原始的。如果它位于$ \ boldsymbol {g} $的阳性亚组$ \ boldsymbol {x} $中，则说明它是不明显的。如果可以选择$ n _ {\ Mathrm {aut} \ boldsymbol {g}}（h）$，则可以选择$ \ boldsymbol {x} $，这是非常不可思议$ \ boldsymbol {g} $。 我们研究了非类别原始简单组$ h $ $ h $的可能嵌入代数组$ \ boldsymbol {g} $在特征coprime到$ | h | $时$（\ boldsymbol {g}，h）$是$（\ boldsymbol {f boldsymbol {f} $（\ boldsymbol {f} _4，\ mathrm {psl} _2（27））$，$（\ boldsymbol {e} _7，\ \ m缩\ Mathrm {psl} _2（29）_2（29））$，$（\ boldsymbol {\ boldsymbol {e} _7} _7，$ mathrmmmathrmmmatrm = 7，特别是，我们在合适的有限字段$ k $上构建了$ h $ in $ g $的$ h $，并用它们来推定$ g $和$ g $和$ \ boldsymbol {g} $的连轭类别的数量，以及$ n _ {\ edrowlline {g}}}（h）$是否是$ subgup of $ subpline $。我们还研究了$ h \ simeq \ mathrm {alt} _6 \ simeq \ simeq \ mathrm {psl} _2（9）$时， $ h $是$ \ boldsymbol {g} $的强烈不可或缺的子组;特别是，$ n _ {\ overline {g}}（h）$不是$ \ overline {g} $的最大子组。
Let $\boldsymbol{G}$ be an algebraic group of exceptional Lie type in characteristic $p$, $G=\boldsymbol{G}^σ$ its fixed-point subgroup under the action of a Steinberg endomorphism $σ$, 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 $\overline{G}$.
