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

Open Mathematics (2012)

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

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topGiovanni 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] 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] Casolo C., On the structure of groups with all subgroups subnormal, J. Group Theory, 2002, 5(3), 293–300 Zbl1002.20016
- [3] Everest G., Ward T., An Introduction to Number Theory, Grad. Texts in Math., 232, Springer, London, 2005 Zbl1089.11001
- [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] 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] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups. I, Ergeb. Math. Grenzgeb., 62, Springer, New York-Berlin, 1972
- [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] 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] Smith H., On non-nilpotent groups with all subgroups subnormal, Ricerche Mat., 2001, 50(2), 217–221 Zbl1097.20512
- [10] Smith H., Groups with all subgroups subnormal or nilpotent-by-Chernikov, Rend. Sem. Mat. Univ. Padova, 126, 2011, 245–253 Zbl1256.20027

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