Dixon, Martyn R.; Evans, Martin J.; Kurdachenko, Leonid A. Linear groups with the minimal condition on subgroups of infinite central dimension. (English) Zbl 1055.20042 J. Algebra 277, No. 1, 172-186 (2004). Let \(V\) be a vector space over a field \(F\). A subgroup \(G\) of the general linear group \(\text{GL}(V)\) on \(V\) is said to have infinite central dimension if \(\dim_F(V/C_V(G))\) is infinite. The authors study subgroups \(G\) of \(\text{GL}(V)\) of infinite central dimension whose set of subgroups with infinite central dimension satisfies the descending chain condition. They have a number of nice results. For example, if \(G\) is also locally finite, then \(G\) is almost soluble. As a second example, if \(G\) is also almost locally soluble, then \(G\) is again almost soluble. Further, in both cases, \(G\) satisfies the minimal condition on normal subgroups, if \(\text{char\,}F=0\) then \(G\) is a Chernikov group and if \(\text{char\,}F\neq 0\), then \(G\) is quite close to being a Chernikov group. Reviewer: B. A. F. Wehrfritz (London) Cited in 1 ReviewCited in 17 Documents MSC: 20H20 Other matrix groups over fields 20F50 Periodic groups; locally finite groups 20F22 Other classes of groups defined by subgroup chains 20E25 Local properties of groups Keywords:general linear groups over infinite dimensional spaces; minimal condition on subgroups; finitary linear groups; locally soluble groups; almost soluble groups; Chernikov groups PDFBibTeX XMLCite \textit{M. R. Dixon} et al., J. Algebra 277, No. 1, 172--186 (2004; Zbl 1055.20042) Full Text: DOI References: [1] Belyaev, V. V., Locally finite groups with Černikov Sylow \(p\)-subgroups, Algebra i Logika. Algebra i Logika, Algebra and Logic, 20, 393-402 (1981), (in Russian); English transl. in · Zbl 0496.20024 [2] Belyaev, V. V., Semisimple periodic groups of finitary transformations, Algebra i Logika. Algebra i Logika, Algebra and Logic, 32, 8-16 (1993), (in Russian); English transl. in · Zbl 0837.20049 [3] Belyaev, V. V., Irreducible periodic groups of finitary transformation, Algebra i Logika. Algebra i Logika, Algebra and Logic, 33, 65-77 (1994), (in Russian); English transl. in · Zbl 0835.20050 [4] Belyaev, V. V., Structure of periodic finitary transformation groups, Algebra i Logika. Algebra i Logika, Algebra and Logic, 33, 195-204 (1994), (in Russian); English transl. in · Zbl 0840.20038 [5] Bruno, B.; Phillips, R. E., A note on groups with nilpotent-by-finite proper subgroups, Arch. Math., 65, 369-374 (1995) · Zbl 0857.20014 [6] Gorenstein, D., Finite Groups (1968), Harper & Row: Harper & Row New York · Zbl 0185.05701 [7] Hartley, B., Fixed points of automorphisms of certain locally finite groups and Chevalley groups, J. London Math. Soc., 37, 421-436 (1988) · Zbl 0619.20018 [8] Kegel, O. H.; Wehrfritz, B. A.F., Locally Finite Groups, North-Holland Math. Library, vol. 3 (1973), North-Holland: North-Holland Amsterdam · Zbl 0195.03804 [9] Meierfrankenfeld, U.; Phillips, R. E.; Puglisi, O., Locally solvable finitary linear groups, J. London Math. Soc. (2), 47, 31-40 (1993) · Zbl 0738.20043 [10] Phillips, R. E., The structure of groups of finitary transformations, J. Algebra, 119, 400-448 (1988) · Zbl 0669.20031 [11] Phillips, R. E., Finitary linear groups: a survey, (Hartley, B.; Seitz, G. M.; Borovik, A. V.; Bryant, R. M., Finite and Locally Finite Groups. Finite and Locally Finite Groups, Istanbul 1994. Finite and Locally Finite Groups. Finite and Locally Finite Groups, Istanbul 1994, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 471 (1995), Kluwer Academic: Kluwer Academic Dordrecht), 111-146 · Zbl 0840.20048 [12] Robinson, D. J.S., Finiteness Conditions and Generalized Soluble Groups, vols. 1 and 2, Ergeb. Math. Grenzgeb., Band 62-63 (1972), Springer-Verlag: Springer-Verlag Berlin [13] Silcock, H. L., Metanilpotent groups satisfying the minimal condition for normal subgroups, Math. Z., 135, 165-173 (1974) · Zbl 0261.20015 [14] Wehrfritz, B. A.F., Infinite Linear Groups, Ergeb. Math. Grenzgeb., Band 76 (1973), Springer-Verlag: Springer-Verlag New York · Zbl 0261.20038 [15] Zaicev, D. I., On solvable subgroups of locally solvable groups, Dokl. Akad. Nauk SSSR. Dokl. Akad. Nauk SSSR, Soviet Math. Dokl., 15, 342-345 (1974), (in Russian); English transl. in · Zbl 0322.20017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.