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$A_{1}$ -TYPE SUBGROUPS CONTAINING REGULAR UNIPOTENT ELEMENTS

Published online by Cambridge University Press:  24 April 2019

TIMOTHY C. BURNESS
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK; t.burness@bristol.ac.uk
DONNA M. TESTERMAN
Affiliation:
Institute of Mathematics, Station 8, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland; donna.testerman@epfl.ch
Corresponding

Abstract

Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic $p>0$ and let $X=\text{PSL}_{2}(p)$ be a subgroup of $G$ containing a regular unipotent element $x$ of $G$ . By a theorem of Testerman, $x$ is contained in a connected subgroup of $G$ of type $A_{1}$ . In this paper we prove that with two exceptions, $X$ itself is contained in such a subgroup (the exceptions arise when $(G,p)=(E_{6},13)$ or $(E_{7},19)$ ). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on $p$ and the embedding of $X$ in $G$ . We discuss applications of our main result to the study of the subgroup structure of finite groups of Lie type.

Type
Research Article
Creative Commons
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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