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 \({\mathcal C}\).

Readership

Graduate students and research mathematicians interested in group theory and generalizations.

Table of Contents

• Introduction
• Labelled diagrams
• Essential embeddings
• Unipotent classes
• Centralizers
• Results
• \((\text {Aut} G)\)-conjugacy
• Tables of \(A_1\) subgroups
• References