On locally graded barely transitive groups

Cansu Betin; Mahmut Kuzucuoğlu

Open Mathematics (2013)

  • Volume: 11, Issue: 7, page 1188-1196
  • ISSN: 2391-5455

Abstract

top
We show that a barely transitive group is totally imprimitive if and only if it is locally graded. Moreover, we obtain the description of a barely transitive group G for the case G has a cyclic subgroup 〈x〉 which intersects non-trivially with all subgroups and for the case a point stabilizer H of G has a subgroup H 1 of finite index in H satisfying the identity χ(H 1) = 1, where χ is a multi-linear commutator of weight w.

How to cite

top

Cansu Betin, and Mahmut Kuzucuoğlu. "On locally graded barely transitive groups." Open Mathematics 11.7 (2013): 1188-1196. <http://eudml.org/doc/269238>.

@article{CansuBetin2013,
abstract = {We show that a barely transitive group is totally imprimitive if and only if it is locally graded. Moreover, we obtain the description of a barely transitive group G for the case G has a cyclic subgroup 〈x〉 which intersects non-trivially with all subgroups and for the case a point stabilizer H of G has a subgroup H 1 of finite index in H satisfying the identity χ(H 1) = 1, where χ is a multi-linear commutator of weight w.},
author = {Cansu Betin, Mahmut Kuzucuoğlu},
journal = {Open Mathematics},
keywords = {Locally graded groups; Locally finite groups; Quasi-finite groups; Splitting automorphism; infinite permutation groups; barely transitive groups; locally graded groups; locally finite groups},
language = {eng},
number = {7},
pages = {1188-1196},
title = {On locally graded barely transitive groups},
url = {http://eudml.org/doc/269238},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Cansu Betin
AU - Mahmut Kuzucuoğlu
TI - On locally graded barely transitive groups
JO - Open Mathematics
PY - 2013
VL - 11
IS - 7
SP - 1188
EP - 1196
AB - We show that a barely transitive group is totally imprimitive if and only if it is locally graded. Moreover, we obtain the description of a barely transitive group G for the case G has a cyclic subgroup 〈x〉 which intersects non-trivially with all subgroups and for the case a point stabilizer H of G has a subgroup H 1 of finite index in H satisfying the identity χ(H 1) = 1, where χ is a multi-linear commutator of weight w.
LA - eng
KW - Locally graded groups; Locally finite groups; Quasi-finite groups; Splitting automorphism; infinite permutation groups; barely transitive groups; locally graded groups; locally finite groups
UR - http://eudml.org/doc/269238
ER -

References

top
  1. [1] Arikan A., On locally graded non-periodic barely transitive groups, Rend. Semin. Mat. Univ. Padova, 2007, 117, 141–146 Zbl1166.20001
  2. [2] Arikan A., Trabelsi N., On certain characterizations of barely transitive groups, Rend. Semin. Mat. Univ. Padova, 2010, 123, 203–210 [WoS] Zbl1202.20002
  3. [3] Belyaev V.V., Groups of Miller-Moreno type, Siberian Math. J., 1978, 19(3), 356–360 http://dx.doi.org/10.1007/BF01875284[Crossref] Zbl0409.20027
  4. [4] Belyaev V.V., Inert subgroups in infinite simple groups, Siberian Math. J., 1993, 34(4), 606–611 http://dx.doi.org/10.1007/BF00975160[Crossref] Zbl0831.20033
  5. [5] Belyaev V.V., Kuzucuoglu M., Locally finite barely transitive groups, Algebra Logic, 2003, 42(3), 147–152 http://dx.doi.org/10.1023/A:1023946008218[Crossref] Zbl1033.20001
  6. [6] Betin C., Kuzucuoğlu M., Description of barely transitive groups with soluble point stabilizer, Comm. Algebra, 2009, 37(6), 1901–1907 http://dx.doi.org/10.1080/00927870802210076[WoS][Crossref] Zbl1181.20002
  7. [7] Bruno B., Phillips R.E., On minimal conditions related to Miller-Moreno type groups, Rend. Sem. Mat. Univ. Padova, 1983, 69, 153–168 Zbl0522.20022
  8. [8] Bhattacharjee M., Macpherson D., Möller R.G., Neumann P.M., Notes on Infinite Permutation Groups, Texts Read. Math., 12, Lecture Notes in Math., 1698, Hindustan Book Agency/Springer, New Delhi/Berlin, 1997 Zbl0916.20001
  9. [9] Dixon J.D., Mortimer B., Permutation Groups, Grad. Texts in Math., 163, Springer, New York, 1996 http://dx.doi.org/10.1007/978-1-4612-0731-3[Crossref] 
  10. [10] Dixon M.R., Evans M.J., Obraztsov V.N., Wiegold J., Groups that are covered by non-abelian simple groups, J. Algebra, 2000, 223(2), 511–526 http://dx.doi.org/10.1006/jabr.1999.8051[Crossref] 
  11. [11] Hartley B., Kuzucuoğlu M., Non-simplicity of locally finite barely transitive groups, Proc. Edinburgh Math. Soc., 1997, 40(3), 483–490 http://dx.doi.org/10.1017/S0013091500023968[Crossref] Zbl0904.20031
  12. [12] Hughes D.R., Thompson J.G., The H-problem and the structure of H-groups, Pacific J. Math., 1959, 9, 1097–1101 http://dx.doi.org/10.2140/pjm.1959.9.1097[Crossref] Zbl0098.25201
  13. [13] Kegel O.H., Wehrfritz B.A.F., Locally Finite Groups, North-Holland Math. Library, 3, North-Holland/Elsevier, Amsterdam-London/New York, 1973 
  14. [14] Khukhro E.I., Nilpotent Groups and Their Automorphisms, de Gruyter Exp. Math., 8, Walter de Gruyter, Berlin, 1993 http://dx.doi.org/10.1515/9783110846218[Crossref] 
  15. [15] Khukhro E.I., Makarenko N.Yu., Large characteristic subgroups satisfying multilinear commutator identities, J. Lond. Math. Soc., 2007, 75(3), 635–646 http://dx.doi.org/10.1112/jlms/jdm047[Crossref] Zbl1132.20013
  16. [16] Kuzucuoğlu M., Barely transitive permutation groups, Arch. Math. (Basel), 1990, 55(6), 521–532 http://dx.doi.org/10.1007/BF01191686[Crossref] Zbl0694.20004
  17. [17] Kuzucuoğlu M., On torsion-free barely transitive groups, Turkish J. Math., 2000, 24(3), 273–276 Zbl0984.20001
  18. [18] Neumann P.M., The lawlessness of groups of finitary permutations, Arch. Math. (Basel), 1975, 26(6), 561–566 http://dx.doi.org/10.1007/BF01229781[Crossref] Zbl0338.20037
  19. [19] Obraztsov V.N., Simple torsion-free groups in which the intersection of any two non-trivial subgroups is non-trivial, J. Algebra, 1998, 199(1), 337–343 http://dx.doi.org/10.1006/jabr.1997.7185[Crossref] 
  20. [20] Ol’shanskii A.Yu., Infinite groups with cyclic subgroups, Soviet Math. Dokl., 1979, 20(2), 343–346 
  21. [21] Ol’shanskii A.Yu., Groups of bounded period with subgroups of prime order, Algebra and Logic, 1982, 21(5), 369–418 http://dx.doi.org/10.1007/BF02027230[Crossref] 
  22. [22] Ol’shanskii A.Yu., Geometry of Defining Relations in Groups, Math. Appl. (Soviet Ser.), 70, Kluwer, Dordrecht, 1991 http://dx.doi.org/10.1007/978-94-011-3618-1[Crossref] 
  23. [23] Robinson D.J.S., A Course in the Theory of Groups, 2nd ed., Grad. Texts in Math., 80, Springer, New York, 1996 http://dx.doi.org/10.1007/978-1-4419-8594-1 
  24. [24] Sozutov A.I., Residually finite groups with nontrivial intersections of pairs of subgroups, Siberian Math. J., 2000, 41(2), 362–365 http://dx.doi.org/10.1007/BF02674606[Crossref] Zbl0956.20018
  25. [25] Tomkinson M.J., FC-Groups, Res. Notes in Math., 96, Pitman, Boston, 1984 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.