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Subring subgroups of symplectic groups in characteristic 2


Authors: A. Bak and A. Stepanov
Original publication: Algebra i Analiz, tom 28 (2016), nomer 4.
Journal: St. Petersburg Math. J. 28 (2017), 465-475
MSC (2010): Primary 14L15
DOI: https://doi.org/10.1090/spmj/1459
Published electronically: May 4, 2017
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Abstract: In 2012, the second author obtained a description of the lattice of subgroups of a Chevalley group $ G(\Phi ,A)$ that contain the elementary subgroup $ E(\Phi ,K)$ over a subring $ K\subseteq A$ provided $ \Phi =B_n,$ $ C_n$, or $ F_4$, $ n\ge 2$, and $ 2$ is invertible in $ K$. It turned out that this lattice is a disjoint union of ``sandwiches'' parametrized by the subrings $ R$ such that $ K\subseteq R\subseteq A$. In the present paper, a similar result is proved in the case where $ \Phi =C_n$, $ n\ge 3$, and $ 2=0$ in $ K$. In this setting, more sandwiches are needed, namely those parametrized by the form rings $ (R,\Lambda )$ such that $ K\subseteq \Lambda \subseteq R\subseteq A$. The result generalizes Ya. N. Nuzhin's theorem of 2013 concerning the root systems $ \Phi =B_n,$ $ C_n$, $ n\ge 3$, where the same description of the subgroup lattice is obtained, but under the condition that $ A$ and $ K$ are fields such that $ A$ is algebraic over $ K$.


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Additional Information

A. Bak
Affiliation: Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany
Email: bak@mathematik.uni-bielefeld.de

A. Stepanov
Affiliation: Mathematics and Mechanics Department, St. Petersburg State University, Universitetskiĭ pr. 28, 198504 St. Petersburg; St. Petersburg Electrotechnical University, Russia
Email: stepanov239@gmail.com

DOI: https://doi.org/10.1090/spmj/1459
Keywords: Symplectic group, commutative ring, subgroup lattice, Bak unitary group, group identity with constants, small unipotent element, nilpotent structure of $K1$
Received by editor(s): February 1, 2016
Published electronically: May 4, 2017
Additional Notes: The second author was supported by Russian Science Foundation, grant no. 14-11-00297
Article copyright: © Copyright 2017 American Mathematical Society