Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov

Giovanni Cutolo; Howard Smith

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 942-949
  • ISSN: 2391-5455

Abstract

top
Let G be a locally finite group satisfying the condition given in the title and suppose that G is not nilpotent-by-Chernikov. It is shown that G has a section S that is not nilpotent-by-Chernikov, where S is either a p-group or a semi-direct product of the additive group A of a locally finite field F by a subgroup K of the multiplicative group of F, where K acts by multiplication on A and generates F as a ring. Non-(nilpotent-by-Chernikov) extensions of this latter kind exist and are described in detail.

How to cite

top

Giovanni Cutolo, and Howard Smith. "Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov." Open Mathematics 10.3 (2012): 942-949. <http://eudml.org/doc/269558>.

@article{GiovanniCutolo2012,
abstract = {Let G be a locally finite group satisfying the condition given in the title and suppose that G is not nilpotent-by-Chernikov. It is shown that G has a section S that is not nilpotent-by-Chernikov, where S is either a p-group or a semi-direct product of the additive group A of a locally finite field F by a subgroup K of the multiplicative group of F, where K acts by multiplication on A and generates F as a ring. Non-(nilpotent-by-Chernikov) extensions of this latter kind exist and are described in detail.},
author = {Giovanni Cutolo, Howard Smith},
journal = {Open Mathematics},
keywords = {Locally finite groups; Subnormal subgroups; Nilpotent-by-Chernikov groups; locally finite groups; subnormal subgroups; nilpotent-by-Chernikov groups},
language = {eng},
number = {3},
pages = {942-949},
title = {Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov},
url = {http://eudml.org/doc/269558},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Giovanni Cutolo
AU - Howard Smith
TI - Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 942
EP - 949
AB - Let G be a locally finite group satisfying the condition given in the title and suppose that G is not nilpotent-by-Chernikov. It is shown that G has a section S that is not nilpotent-by-Chernikov, where S is either a p-group or a semi-direct product of the additive group A of a locally finite field F by a subgroup K of the multiplicative group of F, where K acts by multiplication on A and generates F as a ring. Non-(nilpotent-by-Chernikov) extensions of this latter kind exist and are described in detail.
LA - eng
KW - Locally finite groups; Subnormal subgroups; Nilpotent-by-Chernikov groups; locally finite groups; subnormal subgroups; nilpotent-by-Chernikov groups
UR - http://eudml.org/doc/269558
ER -

References

top
  1. [1] Asar A.O., Locally nilpotent p-groups whose proper subgroups are hypercentral or nilpotent-by-Chernikov, J. London Math. Soc., 2000, 61(2), 412–422 http://dx.doi.org/10.1112/S0024610799008479 Zbl0961.20031
  2. [2] Casolo C., On the structure of groups with all subgroups subnormal, J. Group Theory, 2002, 5(3), 293–300 Zbl1002.20016
  3. [3] Everest G., Ward T., An Introduction to Number Theory, Grad. Texts in Math., 232, Springer, London, 2005 Zbl1089.11001
  4. [4] Möhres W., Auflösbare Gruppen mit endlichem Exponenten, deren Untergruppen alle subnormal sind. I, II, Rend. Sem. Mat. Univ. Padova, 1989, 81, 255–268, 269–287 Zbl0695.20021
  5. [5] Napolitani F., Pegoraro E., On groups with nilpotent by Černikov proper subgroups, Arch. Math. (Basel), 1997, 69(2), 89–94 http://dx.doi.org/10.1007/s000130050097 Zbl0897.20021
  6. [6] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups. I, Ergeb. Math. Grenzgeb., 62, Springer, New York-Berlin, 1972 
  7. [7] Smith H., Hypercentral groups with all subgroups subnormal, Bull. London Math. Soc., 1983, 15(3), 229–234 http://dx.doi.org/10.1112/blms/15.3.229 Zbl0491.20025
  8. [8] Smith H., Groups with all non-nilpotent subgroups subnormal, In: Topics in Infinite Groups, Quad. Mat., 8, Seconda Università degli Studi di Napoli, Caserta, 2001, 309–326 Zbl1017.20018
  9. [9] Smith H., On non-nilpotent groups with all subgroups subnormal, Ricerche Mat., 2001, 50(2), 217–221 Zbl1097.20512
  10. [10] Smith H., Groups with all subgroups subnormal or nilpotent-by-Chernikov, Rend. Sem. Mat. Univ. Padova, 126, 2011, 245–253 Zbl1256.20027

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.