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

• Autorzy: Giovanni Cutolo , Howard Smith
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Locally finite groups; Subnormal subgroups; Nilpotent-by-Chernikov groups

Kategorie tematyczne

• 20F50: Periodic groups; locally finite groups
• 20E15: Chains and lattices of subgroups, subnormal subgroups
• 20F19: Generalizations of solvable and nilpotent groups

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 3
• Strony: 942-949

Daty

wydano

2012-06-01

online

2012-03-24

Twórcy

autor

Giovanni Cutolo

• Università degli Studi di Napoli “Federico II”

autor

Howard Smith

• Bucknell University

Bibliografia

• [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
• [2] Casolo C., On the structure of groups with all subgroups subnormal, J. Group Theory, 2002, 5(3), 293–300
• [3] Everest G., Ward T., An Introduction to Number Theory, Grad. Texts in Math., 232, Springer, London, 2005
• [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
• [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
• [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
• [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
• [9] Smith H., On non-nilpotent groups with all subgroups subnormal, Ricerche Mat., 2001, 50(2), 217–221
• [10] Smith H., Groups with all subgroups subnormal or nilpotent-by-Chernikov, Rend. Sem. Mat. Univ. Padova, 126, 2011, 245–253

Identyfikatory

DOI

10.2478/s11533-012-0020-z