Abstract
This is a survey article outlining the current state of research in the area of finitary linear groups. While many aspects of the subject are covered, special emphasis is placed on foundations, locally solvable groups, and locally finite groups. Generally, proofs are not included.
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References
V. V. Belyaev, Semi simple periodic groups of finitary transformations, Algebra and Logic 32 (1) (1993), 17–33.
V. V. Belyaev, Finitary linear representations of infinite symmetric and alternating groups, Algebra and Logic 32 (6) (1993), 319–327.
V. V. Belyaev, Local characterizations of periodic simple groups of finitary transformations, Algebra and Logic 32 (3) (1993), 231–250.
V. V. Belyaev, The structure of p-groups of finitary transformations, Algebra and Logic 31 (5) (1992), 271–284.
V. V. Belyaev, Local characterizations of infinite alternating and Lie type groups, Algebra and Logic 31 (4) (1992), 221–234.
V. V. Belyaev, Locally finite simple groups as a product of two inert subgroups, Algebra and Logic 31 (4) (1992), 216–221.
V. V. Belyaev, Locally solvable groups of finitary transformations, Algebra and Logic 31 (6) (1992), 331–342.
V. V. Belyaev, Locally finite subgroups with a finite inseparable subgroup, Siberian Math. J. 34 (1993), 218–232.
V. V. Belyaev, Inert subgroups in infinite simple groups, Siberian Math. J. 34 (1993), 606–611.
V. V. Belyaev, Inert subgroups in infinite locally finite groups, these Proceedings.
B. Bruno and R. E. Phillips, Residual properties of finitary linear groups, J. Algebra 166 (1994), 379–392.
P. Cameron, Oligomorphic Permutation Groups, London Math. Soc. Lecture Note Series, 152, Cambridge University Press, Cambridge, New York, 1990.
P. Cameron and J. I. Hall, Some groups generated by transvection subgroups, J. Algebra 140 (1991), 184–209.
C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York, 1962.
J. Dieudonné, Les determinants sur un corps non commutatif, Bull. Math. Soc. France 71 (1943), 27–45.
J. D. Dixon, The Structure of Linear Groups, Van Nostrand Reinhold Co., New York, 1971.
J. I. Hall, Infinite alternating groups as finitary linear transformations groups, J. Algebra 119 (1988), 337–359.
J. I. Hall, Finitary linear groups and elements of finite degree, Arch. Math. 50 (1988), 315–318.
J. I. Hall, Locally finite simple groups of finitary transformations, these Proceedings.
J. I. Hall, The number of trace valued forms and extraspecial groups, J. London Math. Soc.(2) 37 (1988), 1–13.
J. I. Hall, M. Liebeck and G. Seitz, Generators for finite simple groups, with applications to linear groups, Quart. J. Math. 43 (2) (1992), 441–458.
J.I. Hall and B. Hartley, A group theoretical characterization of simple, locally finite, finitary linear groups, Arch. Math. 60 (1993), 108–114.
P. Hall, Wreath powers and characteristically simple groups, Proc. Cambridge Philos. Soc. 58 (1962), 49–71.
B. Hartley, Simple locally finite groups, these Proceedings.
K. Hickin, Some applications of tree limits to groups, Part 1, Trans. Amer. Math. Soc. 305 (2) (1988), 797–839.
K. Hickin and R. E. Phillips, Some countably stunted finitary permutation groups, to appear.
I. D. Ivanjuta, Sylow p-subgroups of the countable symmetric group, Ukrain. Mat. Zh. 15 (1963), 240–249 (in Russian).
I. D. Ivanjuta, On complete sets of Sylow subgroups of a countable symmetric group, Ukrain. Mat. Zh. 18 (1966), 112–115 (in Russian).
E. G. Kosman, Geometrical characterizations of Sylow p-subgroups of a bounded linear group, Ukranian Math. J. 40 (1988), 337–343.
E. G. Kosman, Constructions of Sylow subgroups of a bounded linear group, Ukranian Math. J. 39 (1987), 144–149.
O. H. Kegel and D. Schmidt, Existentially closed finitary linear groups, in Groups St Andrews 1989, eds. C. M. Campbell, E. F. Robertson, Cambridge University Press, LMS Lecture Note Series 159, 1991, pp. 355–362.
O. H. Kegel and B. A. F. Wehrfritz, Locally Finite Groups, North-Holland, Amsterdam, London, 1973.
F. Leinen, Absolute irreducibility for finitary linear groups, to appear in Rend. Sem. Mat. Univ. Padova.
F. Leinen, Irreducible representations of periodic finitary linear groups, to appear in J. Algebra.
F. Leinen, Hypercentral unipotent finitary skew linear groups, Comm. Algebra 22 (3) (1994), 939–949.
F. Leinen and O. Puglisi, Unipotent finitary linear groups, J. London Math. Soc. 48 (2) (1993), 59–76.
D. H. McLain. A characteristically simple group, Proc. Cambridge Phil. Soc. 50 (1954), 641–642.
D. H. McLain, On locally nilpotent groups, Proc. Cambridge Phil. Soc. 52 (1956), 5–11.
D. H. McLain, A Class of Locally Nilpotent Groups, Ph.D. dissertation, Cambridge University, 1956.
D. H. McLain, Finiteness conditions in locally soluble groups, J. London Math. Soc. 34 (1959), 101–107.
U. Meierfrankenfeld, Ascending subgroups of finitary linear groups, to appear in J. London Math. Soc.
U. Meierfrankenfeld, Non finitary locally finite simple groups, these Proceedings.
U. Meierfrankenfeld, A characterization of the natural module for some classical groups, to appear.
U. Meierfrankenfeld, A simple subnormal subgroup for locally finite, finitary linear groups, to appear.
U. Meierfrankenfeld, A note on the cohomology of finitary modules, to appear.
U. Meierfrankenfeld, R. E. Phillips, and O. Puglisi, Locally solvable finitary linear groups, J. London Math. Soc. 47 (1993), 31–40.
A. S. Mikles and R. I. Tyskevic, The transitive subgroups of a bounded symmetric group, Veski Akad. Nauk BSSR, Ser. Fiz-Mat. Nauk 6 (1971), 39–45 (in Russian).
P. M. Neumann, The lawlessness of groups of finitary permutations, Arch. Math. 26 (1975), 561–566.
P. M. Neumann, The structure of finitary permutations groups, Arch. Math. 27 (1976), 3–17.
D. S. Passman, Semiprimitivity of group algebras of locally finite groups, II, to appear.
D. S. Passman, The Jacobson radical of a group ring, Proc. London Math. Soc. 38 (3) (1979), 169–192.
D. S. Passman and A. E. Zalesskiῐ, Semiprimitivity of group algebras of locally finite simple groups, Proc. London Math. Soc. 67 (3) (1993), 243–276.
R. E. Phillips, The structure of groups of finitary transformations, J. Algebra 119 (1988),400–448.
R. E. Phillips, Finitary linear groups generated by elements of small degree, to appear.
R. E. Phillips, On absolutely simple locally finite groups, Rend. Sem. Mat. Padova 79 (1988), 213–220.
R. E. Phillips, Primitive, locally finite, finitary linear groups, to appear.
O. Puglisi, Homomorphic images of finitary linear groups, Arch. Math. 60 (1993), 497–504.
O. Puglisi, Free products of finitary linear groups, to appear.
A. Radford, Residually Finite, Locally Finite, Finitary Groups, Ph.D. thesis, Michigan State University, 1995.
D. J. S. Robinson, Finiteness Conditions and Generalized Solvable Groups, Part I, Springer-Verlag,Berlin, Heidelberg, New York, 1972.
D. J. S. Robinson, Finiteness Conditions and Generalized Solvable Groups, Part II, Springer-Verlag,Berlin, Heidelberg, New York, 1972.
A. Rosenberg, The structure of the infinite general linear group, Ann. Math. 68 (1958), 278–294.
W. R. Scott, Group Theory, Prentice Hall, Englewood Cliffs, 1964.
D. Segal, Normal subgroups of finitary permutation groups, Math. Z. 140 (1974), 81–85.
D. Segal, A note on finitary permutation groups, Arch. Math. 25 (1974), 470–471.
V. N. Serezhkin and A. E. Zalesskiῐ, Finite linear groups generated by reflections, Math. USSR-Izvestia 17 (1981), 477–503.
D. A. Suprenenko, On locally nilpotent subgroups of infinite symmetric groups, Dokl. Akad. Nauk SSSR 167 (1966), 302–304 (in Russian); Soviet Math. Doklady 7 (1966), 392–394.
D. A. Suprenenko, Indecomposibility of transitive subgroups of the group SF(X), Dokl. Akad. Nauk SSSR 186 (1969), 779–780 (in Russian); Soviet Math. Doklady 10 (1969), 674–676.
B. A. F. Wehrfritz, Infinite Linear Groups, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
B. A. F. Wehrfritz, Nilpotence in finitary linear groups, Michigan Math. J. 40 (1993), 419–432.
B. A. F. Wehrfritz, Locally solvable finitary skew linear groups, J. Algebra 160 (1993), 226– 241.
B. A. F. Wehrfritz, Nilpotence in finitary skew linear groups, J. Pure Appl. Algebra 83 (1992), 27–41.
B. A. F. Wehrfritz, Algebras generated by locally nilpotent finitary skew linear groups, J. Pure Appl. Algebra 88 (1993), 305–316.
B. A. F. Wehrfritz, Irreducible locally nilpotent finitary skew linear groups, to appear.
B. A. F. Wehrfritz, A Jordan Holder theorem for finitary linear groups, to appear.
B. A. F. Wehrfritz, Locally soluble primitive finitary skew linear groups, to appear.
B. A. F. Wehrfritz, Generalized soluble primitive skew linear groups, to appear.
J. Wiegold, Groups of finitary permutations, Arch. Math. 25 (1974), 466–469.
H. Wielandt, Unendliche Permutationsgruppen, Lecture Notes: Mathematisches Institut der Universität, Tübingen, 1960.
A. E. Zalesskiῐ, Groups of bounded transformations of a vector space, Dokl. Akad. Nauk BSSR 13 (1969), 485–488 (in Russian).
A. E. Zalesskiῐ, Groups of bounded automorphisms, Dokl. Akad. Nauk BSSR 19 (1975), 681–684 (in Russian).
A. E. Zalesskiῐ The structure of certain classes of matrix groups over skewfields, Sibirsk. Mat. Zh. 8 (1967), 978–988 (in Russian).
A. E. Zalesskiῐ, Solvable subgroups of the multiplicative group of a locally finite algebra, Mat. Sb. 61 (103) (1963), 408–417 (in Russian).
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Phillips, R.E. (1995). Finitary Linear Groups: A Survey. In: Hartley, B., Seitz, G.M., Borovik, A.V., Bryant, R.M. (eds) Finite and Locally Finite Groups. NATO ASI Series, vol 471. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0329-9_5
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DOI: https://doi.org/10.1007/978-94-011-0329-9_5
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