| Brand new. We distribute directly for the publisher. 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 ${\mathcal C}$. |
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