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Trends in infinite dimensional linear groups

Published online by Cambridge University Press:  05 July 2011

By
Martyn R. Dixon ,
Leonid A. Kurdachenko ,
Jose M. Muñoz-Escolano  and
Javier Otal
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Martyn R. Dixon
Affiliation:
University of Alabama, U.S.A
Leonid A. Kurdachenko
Affiliation:
National University of Dnepropetrovsk, Ukraine
Jose M. Muñoz-Escolano
Affiliation:
University of Zaragoza, Spain
Javier Otal
Affiliation:
University of Zaragoza, Spain
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

In this paper we present a short survey discussing some recent results in the theory of infinite dimensional linear groups.

Introduction

In this paper R will denote a ring, G will denote a group and A will be a right RG–module. When R is a field, we shall denote it by F and, of course, A is then also a vector space over F. The group GL(F,A), of all F-automorphisms of A, and its subgroups, are called linear groups. Linear groups have played a very important role in algebra and other branches of mathematics. If dim FA (the dimension of A over F) is finite, n say, then a subgroup G of GL(F, A) is called a finite dimensional linear group. It is well known that in this case GL(F,A) can be identified with the group of all invertible n × n matrices with entries in F. The subject of finite dimensional linear groups is among the most studied branches of mathematics, having been built using the interplay between algebraic, geometrical, combinatorial and other methods. This theory is rich in many interesting and important results.

However, the study of the subgroups of GL(F,A) in the case when A has infinite dimension over F has been much more limited and normally requires some additional restrictions. One natural type of restriction to use here is a finiteness condition. The most fruitful example of such restrictions to date has undoubtedly been that of finitary linear groups.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 271 - 282
Publisher: Cambridge University Press
Print publication year: 2011

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References

[1] R., Baer, Finiteness properties of groups, Duke Math. J. 15 (1948), 1021–1032.Google Scholar
[2] R., Baer, Polyminimaxgruppen, Math. Ann. 175 (1968), 1–43.Google Scholar
[3] J. T., Buckley, J. C., Lennox, B. H., Neumann, H., Smith and J., Wiegold, Groups with all subgroups normal–by–finite, J. Austral. Math. Soc. (Ser. A) 59 (1995), 384–398.Google Scholar
[4] O. Yu., Dashkova, M. R., Dixon and L. A., Kurdachenko, Infinite dimensional linear groups with the restriction on the subgroups of infinite ranks, Izvestiua Gomel University 3(36) (2006), 109–123.Google Scholar
[5] 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 (2004), 172–186.Google Scholar
[6] M. R., Dixon, M. J., Evans and L. A., Kurdachenko, Linear groups with the minimal condition on some infinite dimensional subgroups, Ukrainian Math. J. 57(11) (2005), 1726–1740.Google Scholar
[7] M. R., Dixon and L. A., Kurdachenko, Linear groups with infinite central dimension, in Groups St Andrews 2005, Vol. 1 (C. M., Campbell et al., eds.), London Math. Soc. Lecture Note Ser. 399 (CUP, Cambridge 2007), 306–312.Google Scholar
[8] M. R., Dixon, L. A., Kurdachenko and J., Otal, Linear groups with bounded action, Alg. Colloquium, to appear.
[9] M. R., Dixon, L. A., Kurdachenko and J., Otal, Linear groups with finite dimensional orbits, to appear.
[10] J. I., Hall, Periodic simple groups of finitary linear transformations, Ann. of Math. (2) 163 (2006), 445–498.Google Scholar
[11] L. S., Kazarin and L. A., Kurdachenko, Conditions for finiteness and factorization in infinite groups, Russian Math. Surveys 47 (1992), 81–126.Google Scholar
[12] V. V., Kirichenko, L. A., Kurdachenko and N. V., Polyakov, On certain finitary modules, in Third International Algebraic Conference in Ukraine (Nats. Akad. Nauk Ukr., Inst. Mat. Kiev 2002), 283–296.Google Scholar
[13] L. A., Kurdachenko, J. M., Muñoz-Escolano and J., Otal, Antifinitary linear groups, Forum Mat. 20 (2008), 27–44.Google Scholar
[14] L. A., Kurdachenko, J. M., Muñoz-Escolano and J., Otal, Locally nilpotent linear groups with the weak chain conditions on subgroups of infinite central dimension, Pub. Mat. 52 (2008), 151–169.Google Scholar
[15] L. A., Kurdachenko, J. M., Muñoz-Escolano and J., Otal, Soluble linear groups with some restrictions on their subgroups of infinite central dimension, in Ischia Group Theory 2008, to appear.Google Scholar
[16] L. A., Kurdachenko, J. M., Muñoz-Escolano, J., Otal and N. N., Semko, Locally nilpotent linear groups with restrictions on their subgroups of infinite central dimension, Geom. Ded. 138 (2009), 69–81.Google Scholar
[17] L. A., Kurdachenko, J., Otal and I. Ya., Subbotin, Groups with Prescribed Quotient Groups and Associated Module Theory, Series in Algebra 8 (World Scientific, New Jersey 2002).Google Scholar
[18] L. A., Kurdachenko, J., Otal and I. Ya., Subbotin, Artinian Modules over Group Rings, Frontiers in Mathematics (Birkhäuser, Basel 2007). Dixon et al.: Trends in infinite dimensional linear groups 282Google Scholar
[19] L. A., Kurdachenko, A. V., Sadovnichenko and I. Ya., Subbotin, On some infinite dimensional linear groups, Cent. Eur. J. Math. 8 (2010), 261–265.Google Scholar
[20] L. A., Kurdachenko, I. Ya., Subbotin and V. A., Chupordya, On bounded artinian finitary modules, Internat. J. Algebra Comput. 17(4) (2007), 881–893.Google Scholar
[21] J. C., Lennox and D. J. S., Robinson, The Theory of Infinite Soluble Groups (OUP, Oxford 2004).Google Scholar
[22] J. M., Muñoz-Escolano, J., Otal and N. N., Semko, Periodic linear groups with the weak chain conditions on subgroups of infinite central dimension, Comm. Alg. 36(2) (2008), 749–763.Google Scholar
[23] B. H., Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236–248.Google Scholar
[24] B. H., Neumann, Groups with finite classes of conjugate subgroups, Math. Z. 63 (1955), 76–96.Google Scholar
[25] R. E., Phillips, Finitary linear groups: a survey, in Finite and Locally Finite Groups, Istanbul 1994 (B., Hatley et al., eds.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 471 (Kluwer Acad. Publ., Dordrecht 1995), 111–146.Google Scholar
[26] B. A. F., Wehrfritz, Infinite Linear Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 76, (Springer–Verlag, New York, Heidelberg, Berlin 1973)Google Scholar
[27] B. A. F., Wehrfritz, Finite-finitary groups of automorphisms, J. Algebra App. 1 (2002), 375–389.Google Scholar
[28] B. A. F., Wehrfritz, On generalized finitary groups, J. Algebra 247 (2002), 707–727.Google Scholar
[29] B. A. F., Wehrfritz, Finitary and artinian-finitary groups over the integers ℤ, Ukrainian Math. J. 54 (2002), 753–763.Google Scholar
[30] B. A. F., Wehrfritz, Artinian-finitary groups over commutative rings and non-commutative rings, J. London Math. Soc. 70 (2004), 325–340.Google Scholar
[31] D. I., Zaitsev, The groups satisfying the weak minimal condition, Ukrainian Math. J. 20 (1968), 472–482.Google Scholar

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