| Summary: | Abstract - Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p$. Under some mild restrictions on $p$, we classify all conjugacy classes of closed connected subgroups $X$ of type $A_1$; for each such class of subgroups, we also determine the connected centralizer and the composition factors in the action on the Lie algebra ${{\mathcal L}}(G)$ of $G$. Moreover, we show that ${{\mathcal L}}(C_G(X))=C_{{{{\mathcal L}}(G)}}(X)$ for each subgroup $X$.These results build upon recent work of Liebeck and Seitz, who have provided similar detailed information for closed connected subgroups of rank at least $2$. In addition, for any such subgroup $X$ we identify the unipotent class ${{\mathcal C}}$ meeting it. Liebeck and Seitz proved that the labelled diagram of $X$, obtained by considering the weights in the action of a maximal torus of $X$ on ${{\mathcal L}}(G)$, determines the ($\mathrm{{Aut}}\,G$)-conjugacy class of $X$. We show that in almost all cases the labelled diagram of the class ${{\mathcal C}}$ may easily be obtained from that of $X$; furthermore, if ${{\mathcal C}}$ is a conjugacy class of elements of order $p$, we establish the existence of a subgroup $X$ meeting $${{\mathcal C}}$ and having the same labelled diagram as ${{\mathcal C}}$.
|
|---|
