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Locally graded groups with all subgroups normal-by-finite

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Howard Smith  and
James Wiegold
Howard Smith
Affiliation:
Department of MathematicsBucknell UniversityLewisburg, PA 17837USA e-mail: howsmith@bunknell.edu
James Wiegold
Affiliation:
School of MathematicsUniversity of Wales College of CardiffCardiff CF2 4AG Wales e-mail: smajw@cardiff.ac.uk
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Abstract

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In a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that |H: CoreG (H)| is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that |H: CoreG(H)| ≤ n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

MSC classification

Secondary: 20E26: Residual properties and generalizations; residually finite groups 20F50: Periodic groups; locally finite groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 2 , April 1996 , pp. 222 - 227
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Buckley, J. T., Lennox, John C., Neumann, B. H., Smith, Howard and Wiegold, James, ‘Groups with all subgroups normal-by-finite’, J. Austral. Math. Soc. (Ser.A) 59 (1995), 384–398.CrossRefGoogle Scholar
[2]Fuchs, L., Infinite abelian groups (Academic Press, New York, 1970).Google Scholar
[3]Neumann, B. H., ‘Groups with finite classes of conjugate subgroups’, Math. Z. 63 (1955), 76–96.CrossRefGoogle Scholar
[4]Smith, Howard, ‘On homomorphic images of locally graded groups’, Rend. Sem. Mat. Padova, to appear.Google Scholar
[5]Zel'manov, E. I., ‘Solution of the restricted Burnside problem for groups of odd exponent’, Izv. Akad. Nauk. SSR Ser. Mat. 54 (1990), 42–59.Google Scholar
[6]Zel'manov, E. I., ‘Solution of the restricted Burnside problem for 2-groups’, Mat. Sb. 182 (1991), 568–592.Google Scholar
[7]The Kourovka notebook (Unsolved problems in groups theory) (Russian Academy of Sciences, Novosibirsk, 1992).Google Scholar