# \(A_1\) Subgroups of Exceptional Algebraic Groups

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*R. Lawther; D. M. Testerman*

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 \({\mathcal C}\).