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Locally graded groups with all subgroups normal-by-finite

Published online by Cambridge University Press:  09 April 2009

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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1996

References

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