ON GROUPS WITH TWO ISOMORPHISM CLASSES OF DERIVED SUBGROUPS

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....

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Bibliographic Details
Published inGlasgow mathematical journal Vol. 55; no. 3; pp. 655 - 668
Main Authors LONGOBARDI, PATRIZIA, MAJ, MERCEDE, ROBINSON, DEREK J. S., SMITH, HOWARD
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.09.2013
Subjects
Algebra
Algebraic group theory
Isomorphism
Mathematical analysis
Number theory
Subgroups
Primary 20F14
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ISSN0017-0895
1469-509X
DOI10.1017/S0017089512000821

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Summary: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.
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ISSN:0017-0895
1469-509X
DOI:10.1017/S0017089512000821