On some infinite dimensional linear groups

Leonid Kurdachenko; Alexey Sadovnichenko; Igor Subbotin

Open Mathematics (2009)

  • Volume: 7, Issue: 2, page 176-185
  • ISSN: 2391-5455

Abstract

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Let F be a field, A be a vector space over F, and GL(F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dimF(B/CoreG(B)) is finite. In the present article we begin the study of subgroups G of GL(F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = charF.

How to cite

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Leonid Kurdachenko, Alexey Sadovnichenko, and Igor Subbotin. "On some infinite dimensional linear groups." Open Mathematics 7.2 (2009): 176-185. <http://eudml.org/doc/269113>.

@article{LeonidKurdachenko2009,
abstract = {Let F be a field, A be a vector space over F, and GL(F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dimF(B/CoreG(B)) is finite. In the present article we begin the study of subgroups G of GL(F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = charF.},
author = {Leonid Kurdachenko, Alexey Sadovnichenko, Igor Subbotin},
journal = {Open Mathematics},
keywords = {Vector space; Linear groups; Periodic groups; Invariant subspace; infinite-dimensional linear groups; periodic groups; soluble groups},
language = {eng},
number = {2},
pages = {176-185},
title = {On some infinite dimensional linear groups},
url = {http://eudml.org/doc/269113},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Leonid Kurdachenko
AU - Alexey Sadovnichenko
AU - Igor Subbotin
TI - On some infinite dimensional linear groups
JO - Open Mathematics
PY - 2009
VL - 7
IS - 2
SP - 176
EP - 185
AB - Let F be a field, A be a vector space over F, and GL(F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dimF(B/CoreG(B)) is finite. In the present article we begin the study of subgroups G of GL(F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = charF.
LA - eng
KW - Vector space; Linear groups; Periodic groups; Invariant subspace; infinite-dimensional linear groups; periodic groups; soluble groups
UR - http://eudml.org/doc/269113
ER -

References

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  1. [1] Buckley J.T., Lennox J.C., Neumann B.H., Smith H., Wiegold J., Groups with all subgroups normal-by-finite, J. Austral. Math. Soc. Ser. A, 1995, 59, 384–398 http://dx.doi.org/10.1017/S1446788700037289[Crossref] Zbl0853.20023
  2. [2] Dixon M.R., Evans M.J., Kurdachenko L.A., Linear groups with the minimal condition on subgroups of infinite central dimension, J. Algebra, 2004, 277, 172–186 http://dx.doi.org/10.1016/j.jalgebra.2004.02.029[WoS][Crossref] 
  3. [3] Kurdachenko L.A., Muñoz-Escolano J.M., Otal J., Locally nilpotent linear groups with the weak chain conditions on subgroups of infinite central dimension, Publ. Mat., 2008, 52, 151–169 Zbl1149.20030
  4. [4] Kurdachenko L.A., Otal J,. Subbotin I.Ya., Groups with prescribed quotient groups and associated module theory, World Scientific, New Jersey, 2002 Zbl1019.20001
  5. [5] Kurdachenko L.A, Otal J., Subbotin I.Ya., Artinian modules over group rings, Frontiers in Mathematics, Birkhäuser, Basel, 2007 Zbl1110.16001
  6. [6] Kurdachenko L.A., Subbotin I.Ya., Linear groups with the maximal condition on subgroups of infinite central dimension, Publ. Mat., 2006, 50, 103–131 
  7. [7] Muñoz-Escolano J.M., Otal J., Semko N.N., Periodic linear groups with the weak chain conditions on subgroups of infinite central dimension, Comm. Algebra, 2008, 36, 749–763 http://dx.doi.org/10.1080/00927870701724318[WoS][Crossref] Zbl1141.20030
  8. [8] Neumann B.H., Groups with finite classes of conjugate subgroups, Math. Z, 1955, 63, 76–96 http://dx.doi.org/10.1007/BF01187925[Crossref] Zbl0064.25201
  9. [9] Phillips R.E., Finitary linear groups: a survey, In: Finite and locally finite groups (Istanbul 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 471, Kluwer Acad. Publ., Dordrecht, 1995, 111–146 Zbl0840.20048
  10. [10] Wehrfritz B.A.F., Infinite linear groups, Springer, Berlin, 1973 

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