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ON GROUPS WITH TWO ISOMORPHISM CLASSES OF DERIVED SUBGROUPS

Published online by Cambridge University Press:  25 February 2013

PATRIZIA LONGOBARDI ,
MERCEDE MAJ ,
DEREK J. S. ROBINSON  and
HOWARD SMITH
PATRIZIA LONGOBARDI
Affiliation:
Dipartimento di Matematica, Università di Salerno84084 Fisciano (Salerno), Italy e-mail: plongobardi@unisa.it
MERCEDE MAJ
Affiliation:
Dipartimento di Matematica, Università di Salerno84084 Fisciano (Salerno), Italy e-mail: mmaj@unisa.it
DEREK J. S. ROBINSON
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign Urbana, IL 61801, USA e-mail: dsrobins@illinois.edu
HOWARD SMITH
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA e-mail: howsmith@bucknell.edu
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Abstract

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The structure of groups which have at most two isomorphism classes of derived subgroups ($\mathfrak{D}$2-groups) is investigated. A complete description of $\mathfrak{D}$2-groups is obtained in the case where the derived subgroup is finite: the solution leads an interesting number theoretic problem. In addition, detailed information is obtained about soluble $\mathfrak{D}$2-groups, especially those with finite rank, where algebraic number fields play an important role. Also, detailed structural information about insoluble $\mathfrak{D}$2-groups is found, and the locally free $\mathfrak{D}$2-groups are characterized.

Keywords

Primary 20F14

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 3 , September 2013 , pp. 655 - 668
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

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