Abstract
We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A3-groups and all of whose proper subgroups, which are not soluble A3-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A3-groups and all of whose proper subgroups, which are not soluble A3-groups, have finite fundamental dimension.
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References
R. E. Phillips, “The structure of groups of finitary transformations,” J. Alg., 119,No. 2, 400–448 (1988).
R. E. Philips, “Finitary linear groups: a survey,” in Finite and Locally Finite Groups, B. Hartley et al. (eds.), NATO ASI Ser., Ser. C, Math. Phys. Sci., 471, Kluwer, Dordrecht (1995), pp. 111–146.
M. R. Dixon, M. J. Evans, and L. A. Kurdachenko, “Linear groups with the minimal condition on subgroups of infinite central dimension,” J. Alg., 277,No. 1, 172–186 (2004).
L. A. Kurdachenko and I. Ya. Subbotin, “Linear groups with the maximal condition on subgroups of infinite central dimension,” Publ. Mat., 50,No. 1, 103–131 (2006).
O. Yu. Dashkova, M. R. Dixon, and L. A. Kurdachenko, “Linear groups with rank restrictions on the subgroups of infinite central dimension,” J. Pure Appl. Alg., 208,No. 3, 785–795 (2007).
M. R. Dixon, L. A. Kurdachenko, and O. Yu. Dashkova, “Infinite-dimensional linear groups with restrictions on subgroups of infinite ranks,” Izv. Gomel State Univ., 36,No 3, 109–123 (2006).
A. I. Mal’tsev, “Some classes of infinite soluble groups,” Mat. Sb., 28,No. 2, 567–588 (1951).
D. I. Zaitsev, “Toward a theory of minimax groups,” Ukr. Mat. Zh., 23,No. 5, 652–660 (1971).
V. S. Charin, “Locally soluble groups of finite rank,” Mat. Sb., 41,No. 1, 37–48 (1957).
O. H. Kegel and B. A. Wehrfritz, Locally Finite Groups, North-Holland Math. Library, 3, Elsevier, New York (1973).
M. I. Kargapolov, Selected Writings. Groups [in Russian], Nauka, Novosibirsk (1991), pp. 33–56.
S. N. Černikov, “Infinite locally finite groups with finite Sylow subgroups,” Mat. Sb., 52,No. 1, 647–652 (1960).
N. N. Myagkova, “Groups of finite rank,” Izv. Akad. Nauk SSSR, 13,No. 6, 495–512 (1949).
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Translated from Algebra i Logika, Vol. 46, No. 5, pp. 548–559, September–October, 2007.
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Dashkova, O.Y. Infinite-dimensional linear groups with restrictions on subgroups that are not soluble A 3-groups. Algebra Logic 46, 297–302 (2007). https://doi.org/10.1007/s10469-007-0030-2
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DOI: https://doi.org/10.1007/s10469-007-0030-2