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On the isomorphisms between certain congruence groups


Author: Robert E. Solazzi
Journal: Proc. Amer. Math. Soc. 35 (1972), 405-410
MSC: Primary 20H15
DOI: https://doi.org/10.1090/S0002-9939-1972-0308293-9
MathSciNet review: 0308293
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Abstract: In this paper we study the possible isomorphisms between the congruence subgroups of the classical groups over integral domains by applying the involution-free techniques previously used by O'Meara and the author. We prove, in dimensions at least 6 and characteristic not 2, that a linear congruence group is never isomorphic to a symplectic congruence group nor to an isotropic unitary congruence group whose associated hermitian space has Witt index at least 3.


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  • [1] H. Bass, J. Milnor and J.-P. Serre, Solution of the congruence subgroup problem for $ \mathrm{SL}_n(n \geqq 3)$ and $ \mathrm{Sp}_{2n}(n \geqq 2)$, Inst. Hautes Études Sci. Publ. Math. No. 33 (1967), 59-137. MR 39 #5574. MR 0244257 (39:5574)
  • [2] J. Dieudonné, La géométrie des groupes classiques, 2ième éd., Springer-Verlag, Berlin, 1963. MR 28 #1239. MR 0158011 (28:1239)
  • [3] -, On the automorphisms of the classical groups, Mem. Amer. Math. Soc. No. 2 (1951). MR 13, 531.
  • [4] O. T. O'Meara and H. Zassenhaus, The automorphisms of the linear congruence groups over Dedekind domains, J. Number Theory 1 (1969), 211-221. MR 39 #4292. MR 0242965 (39:4292)
  • [5] O. T. O'Meara, Group-theoretic characterization of transvections using CDC, Math. Z. 110 (1969), 385-394. MR 40 #1486. MR 0248234 (40:1486)
  • [6] C. E. Rickart, Isomorphic groups of linear transformations, Amer. J. Math. 72 (1950), 451-464. MR 11, 729. MR 0035399 (11:729f)
  • [7] -, Isomorphic groups of linear transformations. II, Amer. J. Math. 73 (1951), 697-716. MR 13, 532. MR 0045126 (13:532a)
  • [8] R. E. Solazzi, The automorphisms of certain subgroups of $ \mathrm{PGL}_n(V)$, Illinois J. Math. 16 (1972), 330-349. MR 0291303 (45:397)
  • [9] -, The automorphisms of the unitary groups and their congruence subgroups, Illinois J. Math. (to appear). MR 0430093 (55:3100)
  • [10] -, Isomorphism theory of congruence groups, Bull. Amer. Math. Soc. 77 (1971), 164-168. MR 0294515 (45:3585)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0308293-9
Keywords: Classical group, centralizer, skew-hermitian form, linear, symplectic, unitary congruence group, transvection, isomorphism
Article copyright: © Copyright 1972 American Mathematical Society

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