We study a \( \mathbb{Z}G \)-module A such that \( \mathbb{Z} \) is the ring of integer numbers, the group G has an infinite sectional p-rank (or an infinite 0-rank), C G (A) = 1, A is not a minimax \( \mathbb{Z} \)-module, and, for any proper subgroup H of infinite sectional p-rank (or infinite 0-rank, respectively), the quotient module A/C A (H) is a minimax \( \mathbb{Z} \)-module. It is shown that if the group G is locally soluble, then it is soluble. Some properties of soluble groups of this kind are discussed.
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References
R. E. Phillips, “The structure of groups of finitary transformations,” J. Algebra, 119, No. 2, 400–448 (1988).
R. E. Phillips, “Finitary linear groups: a survey. ‘Finite and locally finite groups’,” NATO ASI., Ser. C. Math. Phys. Sci., 471, 111–146 (1995).
M. R. Dixon, M. J. Evans, and L. A. Kurdachenko, “Linear groups with the minimal condition on subgroups of infinite central dimension,” J. Algebra, 277, No. 1, 172–186 (2004).
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. Algebra, 208, No. 3, 785–795 (2007).
R. Baer and H. Heineken, “Radical groups of finite Abelian subgroup rank,” Ill. J. Math., 16, No. 4, 533–580 (1972).
A. I. Mal’tsev, “On groups of finite rank,” Mat. Sb., 22, No. 2, 351–352 (1948).
L. A. Kurdachenko, “On groups with minimax classes of conjugate elements,” in: Infinite Groups and Related Algebraic Structures [in Russian], Kiev (1993), pp. 160–177.
L. A. Kurdachenko, I. Ya. Subbotin, and N. N. Semko, Insight into Modules over Dedekind Domains, Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2008).
O. Yu. Dashkova, “On modules over group rings of locally soluble groups with rank restrictions on some systems of subgroups,” Asian-Eur. J. Math., 3, No. 1, 45–55 (2010).
O. Yu. Dashkova, “On one class of modules close to Noetherian,” Fund. Prikl. Mat., 15, No. 7, 113–125 (2009).
L. A. Kurdachenko, J. Otal, and I. Ya. Subbotin, Artinian Modules over Group Rings, Birkhäuser, Basel (2007).
B. A. F. Wehrfritz, Infinite Linear Groups, Springer, Berlin (1973).
O. H. Kegel and B. A. F. Wehrfritz, Locally Finite Groups, North-Holland, Amsterdam (1973).
S. Franciosi and F. De Giovanni, “The Shur property and groups with uniform conjugacy classes,” J. Algebra, 174, No. 3, 823–847 (1995).
D. J. R. Robinson, Finiteness Conditions and Generalized Soluble Groups, Vols. 1, 2, Springer, Berlin (1972).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 9, pp. 1206–1217, September, 2011.
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Dashkova, O.Y. On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups. Ukr Math J 63, 1379–1389 (2012). https://doi.org/10.1007/s11253-012-0585-5
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DOI: https://doi.org/10.1007/s11253-012-0585-5