On totally inert simple groups

Martyn Dixon; Martin Evans; Antonio Tortora

Open Mathematics (2010)

  • Volume: 8, Issue: 1, page 22-25
  • ISSN: 2391-5455

Abstract

top
A subgroup H of a group G is inert if |H: H ∩ H g| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.

How to cite

top

Martyn Dixon, Martin Evans, and Antonio Tortora. "On totally inert simple groups." Open Mathematics 8.1 (2010): 22-25. <http://eudml.org/doc/269299>.

@article{MartynDixon2010,
abstract = {A subgroup H of a group G is inert if |H: H ∩ H g| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.},
author = {Martyn Dixon, Martin Evans, Antonio Tortora},
journal = {Open Mathematics},
keywords = {Inert; Totally inert; Simple group; totally inert groups; infinite simple groups; minimal normal subgroups; subgroups of finite index; periodic groups},
language = {eng},
number = {1},
pages = {22-25},
title = {On totally inert simple groups},
url = {http://eudml.org/doc/269299},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Martyn Dixon
AU - Martin Evans
AU - Antonio Tortora
TI - On totally inert simple groups
JO - Open Mathematics
PY - 2010
VL - 8
IS - 1
SP - 22
EP - 25
AB - A subgroup H of a group G is inert if |H: H ∩ H g| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.
LA - eng
KW - Inert; Totally inert; Simple group; totally inert groups; infinite simple groups; minimal normal subgroups; subgroups of finite index; periodic groups
UR - http://eudml.org/doc/269299
ER -

References

top
  1. [1] Belyaev V.V., Inert subgroups in infinite simple groups, Sibirsk. Mat. Zh., 1939, 34(4), 17–23 (in Russian), English translation: Siberian Math. J., 1993, 34(4), 606–611 
  2. [2] Belyaev V.V., Locally finite groups containing a finite inseparable subgroup, Sibirsk. Mat. Zh., 1993, 34, 23–41 (in Russian), English translation: Siberian Math. J., 1993, 34, 218–232 
  3. [3] Belyaev V.V., Inert subgroups in simple locally finite groups, In: Finite and locally finite groups, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 471, 213–218, Kluwer Acad. Publ., Dordrecht, 1995 Zbl0838.20033
  4. [4] Belyaev V.V., Kuzucuoǧlu M., Seçkin E., Totally inert groups, Rend. Sem. Mat. Univ. Padova, 1999, 102, 151–156 Zbl0945.20022
  5. [5] Bergman G. M., Lenstra H. W. Jr., Subgroups close to normal subgroups, J. Algebra, 1989, 127(1), 80–97 http://dx.doi.org/10.1016/0021-8693(89)90275-5 
  6. [6] Dixon M.R., Evans M. J., Smith H., Embedding groups in locally (soluble-by-finite) simple groups, J. Group Theory, 2006, 9, 383–395 http://dx.doi.org/10.1515/JGT.2006.026 Zbl1120.20030
  7. [7] Kegel O.H., Wehrfritz B.A.F., Locally finite groups, North Holland, Amsterdam, 1973 
  8. [8] Neumann H., Varieties of groups, Springer-Verlag, NewYork, 1967 
  9. [9] Olshanskii A.Yu., An infinite group with subgroups of prime orders, Izv. Akad. Nauk SSSR Ser. Mat., 1980, 44(2), 309–321 
  10. [10] Robinson D.J.S., Finiteness conditions and generalized soluble groups, Springer-Verlag, 1972 Zbl0243.20032
  11. [11] Robinson D.J.S., A course in the theory of groups, 2nd edition, Springer-Verlag, NewYork, 1996 
  12. [12] Robinson D.J.S., On inert subgroups of a group, Rend. Sem. Mat. Univ. Padova, 2006, 115, 137–159 Zbl1167.20319
  13. [13] Zel’manov E.I., Solution of the restricted Burnside’s problem for groups of odd exponent, Math. USSR-Izv., 1991, 36(1), 41–60 http://dx.doi.org/10.1070/IM1991v036n01ABEH001946 
  14. [14] Zel’manov E.I., Solution of the restricted Burnside problem for 2-groups, Mat. Sb., 1991, 182(4), 568–592 (in Russian), English translation: Math. USSR-Sb., 1992, 72(2), 543–565 Zbl0752.20017

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.