Let R be an associative ring with 1 and G = GL(n, R) the general linear group of degree n ≥ 3 over R. A goal of the paper is to calculate the relative centralizers of the relative elementary subgroups or the principal congruence subgroups, corresponding to an ideal A ⊴ R modulo the relative elementary subgroups or the principal congruence subgroups, corresponding to another ideal B ⊴ R. Modulo congruence subgroups, the results are essentially easy exercises in linear algebra. But modulo the elementary subgroups, they turned out to be quite tricky, and definitive answers are obtained only over commutative rings or, in some cases, only over Dedekind rings/Dedekind rings of arithmetic type.
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A. S. Ananyevskiy, N. A. Vavilov, and S. S. Sinchuk, “Overgroups of E(m,R) ⊗ E(n,R). I. Levels and normalizers,” St. Petersburg Math. J., 23, No. 5, 819–849 (2015).
A. Bak, “Non-Abelian K-theory: The nilpotent class of K1 and general stability,” KTheory, 4, 363–397 (1991).
A. Bak, R. Hazrat, and N. A. Vavilov, “Localization-completion strikes again: relative K1 is nilpotent by Abelian,” J. Pure Appl. Algebra, 213, 1075–1085 (2009).
A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloq., 7, No. 2, 159–196 (2000).
H. Bass, “K-theory and stable algebra,” Inst. Hautes Études Sci. Publ. Math., No. 22, 5–60 (2002).
H. Bass, J. Milnor, and J.-P. Serre, “Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2),” Publ. Math. Inst. Hautes Etudes Sci., 33, 59–137 (1967).
Z. I. Borewicz and N. A. Vavilov, “Subgroups of the general linear group over a semilocal ring, containing the group of diagonal matrices,” Proc. Steklov. Inst. Math., 148, 41–54 (1980).
Z. I. Borewicz and N. A. Vavilov, “The distribution of subgroups in the full linear group over a commutative ring,” Proc. Steklov Inst. Math., 165, 27–46 (1985).
J. L. Bueso, J. Gómez-Torrencillas, and A. Verschoren, Algorithmic methods in noncommutative algebra: applications to finite groups, Springer Verlag, Berlin et al. (2013).
A. J. Hahn and O. T. O’Meara, The classical groups and K-theory, Springer, Berlin et al. (1989).
R. Hazrat, “Dimension theory and non-stable K1 of quadratic module,” K-Theory, 27, 293–327 (2002).
R. Hazrat, A. Stepanov, N. Vavilov, and Zuhong Zhang, “The yoga of commutators,” J. Math. Sci., 179, No. 6, 662–678 (2011).
R. Hazrat, A. Stepanov, N. Vavilov, and Zuhong Zhang, “Commutators width in Chevalley groups,” Note Matem., 33, No. 1, 139–170 (2013).
R. Hazrat and N. Vavilov, “K1 of Chevalley groups are nilpotent,” J. Pure Appl. Algebra, 179, No. 1, 99–116 (2003).
R. Hazrat and N. Vavilov, “Bak’s work on the K-theory of rings,” J. K-Theory, 4, No. 1, 1–65 (2009).
R. Hazrat, N. Vavilov, and Zuhong Zhang, “The commutators of classical groups,” J. Math. Sci., 222, No. 4, 466–515 (2017).
R. Hazrat and Zuhong Zhang, “Generalized commutator formula,” Commun. Algebra, 39, No. 4, 1441–1454 (2011).
R. Hazrat and Zuhong Zhang, “Multiple commutator formula,” Israel J. Math., 195, 481–505 (2013).
W. van der Kallen, “A module structure on certain orbit sets of unimodular rows,” J. Pure Appl. Algebra, 57, No. 3, 281–316 (1989).
H. Marubayashi, “Primary ideal representations in non-commutative rings,” Math. J. Okayama Univ., 13, No. 1, 1–7 (1967).
A. W. Mason, “On subgroups of GL(n,A) which are generated by commutators. II,” J. Reine Angew. Math., 322, 118–135 (1981).
A. W. Mason, “On non-normal subgroups of GLn(A) which are normalized by elementary matrices,” Illinois J. Math., 28, No. 1, 125–138 (1984).
A. W. Mason and W. W. Stothers, “On subgroups of GL(n,A) which are generated by commutators,” Invent. Math., 23, 327–346 (1974).
D. C. Murdoch, “Contribution to non-commutative ideal theory,” Canad. J. Math., 4, No. 1, 43–57 (1952).
O. Steinfeld, “On ideal-quotients and prime ideals,” Acta Math. Acad. Sci. Hungar., 4, No. 3–4, 289–298 (1953).
L. N. Vaserstein, “On the normal subgroups of the GLn of a ring,” in: Algebraic KTheory, Evanston 1980, Lecture Notes in Math., vol. 854, Springer, Berlin et al. (1981), pp. 454–465.
N. Vavilov, “Unrelativised standard commutator formula,” Zap. Nauchn. Semin. POMI, 470, 38–49 (2018).
N. Vavilov, “Commutators of congruence subgroups in the arithmetic case,” J. Math. Sci., 479, 5–22 (2019).
N. Vavilov, “Towards the reverse decomposition of unipotents. II. The relative case,” J. Math. Sci., 484, 5–23 (2019).
N. A. Vavilov and M. R. Gavrilovich, “A2-proof of structure theorems for Chevalley groups of types E6 and E7,” St. Petersburg Math. J., 16, No. 4, 649–672 (2005).
N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of Chevalley groups of type E6,” St. Petersburg Math. J., 19, No. 5, 699–718 (2008).
N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of Chevalley groups of type E7,” St. Petersburg Math. J., 27, No. 6, 899–921 (2016).
N. A. Vavilov and S. I. Nikolenko, “A2-proof of structure theorems for the Chevalley group of type F4,” St. Petersburg Math. J., 20, No. 4, 527–551 (2009).
N. A. Vavilov and V. A. Petrov, “Overgroups of elementary symplectic groups,” St. Petersburg Math. J., 15, No. 4, 515–543 (2004).
N. A. Vavilov and V. A. Petrov, “Overgroups of EO(n,R),” St. Petersburg Math. J., 19, No. 2, 167–195 (2008).
N. A. Vavilov, and A. V. Stepanov, “Standard commutator formula,” Vestn. St. Petersburg State Univ., ser. 1, 41, No. 1, 5–8 (2008).
N. A. Vavilov, and A. V. Stepanov, “Standard commutator formula, revisited,” Vestn. St.Petersburg State Univ., ser. 1, 43, No. 1, 12–17 (2010).
N. Vavilov and Z. Zhang, “Generation of relative commutator subgroups in Chevalley groups. II,” Proc. Edinburgh Math. Soc., 63, 497–511 (2020).
N. Vavilov and Z. Zhang, “Commutators of relative and unrelative elementary groups, revisited,” J. Math. Sci., 485, 58–71 (2019).
N. Vavilov and Z. Zhang, “Multiple commutators of elementary subgroups: end of the line,” Linear Algebra Applic., 599, 1–17 (2020).
N. Vavilov and Z. Zhang, “Inclusions among commutators of elementary subgroups,” submitted to J. Algebra, 1–26 (2019).
N. Vavilov and Z. Zhang, “Commutators of relative and unrelative elementary subgroups in Chevalley groups,” submitted to Edinburg Math. Soc., 1–18 (2020).
N. Vavilov and Z. Zhang, “Commutators of relative and unrelative elementary unitary groups,” submitted to J. Algebra Number Theory, 1–40 (2020).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 492, 2020, pp. 10–24.
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Vavilov, N.A., Zhang, Z. Relative Centralizers of Relative Subgroups. J Math Sci 264, 4–14 (2022). https://doi.org/10.1007/s10958-022-05973-y
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DOI: https://doi.org/10.1007/s10958-022-05973-y