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

2017/09/20 by Alexander Shchegolev, Shchegolev, Alexander
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1709.07038

openalex publication_date 2017/09/20 · openalex created_date 2022/09/26 · openalex updated_date 2026/07/28

Abstract

In this paper we prove a sandwich classification theorem for subgroups of the\nclassical symplectic group over an arbitrary commutative ring R that contain\nthe elementary block-diagonal (or subsystem) subgroup \Ep(\ν,\nR) corresponding to a unitary equivalence realation \ν such that all\nself-conjugate equivalence classes of \ν are of size at least 4 and all\nnot-self-conjugate classes of \ν are of size at least 5. Namely, given a\nsubgroup H of \Sp(2n, R) such that \Ep(\ν, R)\n\≤ H we show that there exists a unique exact major form net of ideals\n(\σ, \Γ) over R such that \Ep(\σ, \Γ) \≤ H\n\≤ N_\Sp(2n,R)(\Sp(\σ, \Γ)). Further,\nwe describe the normalizer\nN_\Sp(2n,R)(\Sp(\σ, \Γ)) in terms of\ncongruences.\n

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