Abstract
We consider the Sylow p-subgroups, obtained by completion, of the restricted linear group of a countable-dimensional vector space of countable cardinality over a finite field of characteristic p. The geometric approach of O'Meara is used to describe the isomorphisms of linear groups that are not rich or even sufficiently rich in transvections. We prove that between two isomorphic Sylow p-subgroups there is an isomorphism of standard form induced by some locally inner automorphism of the restricted linear group.
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Literature cited
E. G. Kosman, “Construction of Sylow p-subgroups of the restricted linear group,” Ukr. Mat. Zh.,39, No. 2, 173–179 (1987).
E. G. Kosman, “On a geometric characterization of Sylow p-subgroups of the restricted linear group,” Ukr. Mat. Zh.,40, No. 3, 391–397 (1988).
E. G. Kosman, “On the normal structure of Sylow p-subgroups of the restricted linear group,” Ukr. Mat. Zh.,41, No. 10, 1399–1403 (1989).
E. G. Kosman, “Isomorphisms of Sylow p-subgroups of the restricted linear group. I. Preservation of transvections,” Ukr. Mat. Zh.,41, No. 12, 1649–1653 (1989).
O. T. O'Meara, “General theory of isomorphisms of linear groups,” in: Isomorphisms of Classical Groups over Integral Rings [Russian translation], Mir, Moscow (1980), pp. 58–118.
O. T. O'Meara, “Lectures on linear groups,” in: Automorphisms of Classical Groups [Russian translation], Mir, Moscow (1976), pp. 57–167.
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 418–421, March, 1990.
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Kosman, E.G. Isomorphisms of Sylow p-subgroups of the restricted linear group II. Main theorem. Ukr Math J 42, 372–374 (1990). https://doi.org/10.1007/BF01057028
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DOI: https://doi.org/10.1007/BF01057028