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$$A_1$$ Subgroups of Exceptional Algebraic Groups
R. Lawther, Lancaster University, England, and D. M. Testerman, University of Warwick, Coventry, England
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Memoirs of the American Mathematical Society
1999; 131 pp; softcover
Volume: 141
ISBN-10: 0-8218-1966-6
ISBN-13: 978-0-8218-1966-1
List Price: US$51 Individual Members: US$30.60
Institutional Members: US\$40.80
Order Code: MEMO/141/674

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 $${\mathcal C}$$.

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

• $$(\text {Aut} G)$$-conjugacy
• Tables of $$A_1$$ subgroups