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St. Petersburg Mathematical Journal

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Overgroups of elementary block-diagonal subgroups in the classical symplectic group over an arbitrary commutative ring


Author: A. V. Shchegolev
Translated by: the author
Original publication: Algebra i Analiz, tom 30 (2018), nomer 6.
Journal: St. Petersburg Math. J. 30 (2019), 1007-1041
MSC (2010): Primary 20G35
DOI: https://doi.org/10.1090/spmj/1580
Published electronically: September 16, 2019
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Abstract: The paper provides a sandwich classification theorem for subgroups of the classical symplectic group over an arbitrary commutative ring $ R$ that contain the elementary block-diagonal (or subsystem) subgroup $ \mathrm {Ep}(\nu , R)$ corresponding to a unitary equivalence relation $ \nu $ such that all selfconjugate equivalence classes of $ \nu $ are of size at least 4 and all nonselfconjugate classes of $ \nu $ are of size at least 5. Namely, given a subgroup $ H \ge \mathrm {Ep}(\nu , R)$ of $ \mathrm {Sp}(2n, R)$, it is shown that there exists a unique exact major form net of ideals $ (\sigma , \Gamma )$ over $ R$ such that $ \mathrm {Ep}(\sigma , \Gamma ) \le H \le \mathrm {N}_{\operatorname {Sp}(2n,R)}(\mathrm {Sp}(\sigma , \Gamma ))$. Next, the normalizer $ \mathrm {N}_{\mathrm {Sp}(2n,R)}(\mathrm {Sp}(\sigma , \Gamma ))$ is described in terms of congruences.


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Additional Information

A. V. Shchegolev
Affiliation: St. Petersburg State University
Email: a.shchegolev@spbu.ru

DOI: https://doi.org/10.1090/spmj/1580
Keywords: Symplectic group, elementary subgroup, subgroup structure, standard automorphisms, block-diagonal subgroup, localization methods
Received by editor(s): September 24, 2017
Published electronically: September 16, 2019
Additional Notes: This research is supported by the Russian Science Foundation grant no. 17-11-01261
Article copyright: © Copyright 2019 American Mathematical Society