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}$2-groups is obtained in the case where the derived subgroup is finite: the solution leads an interesting number theoretic problem....
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| Published in | Glasgow mathematical journal Vol. 55; no. 3; pp. 655 - 668 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.09.2013
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0017-0895 1469-509X |
| DOI | 10.1017/S0017089512000821 |
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| Abstract | The structure of groups which have at most two isomorphism classes of derived subgroups ($\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}$2-groups are characterized. |
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| AbstractList | The structure of groups which have at most two isomorphism classes of derived subgroups ($\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. Abstract The structure of groups which have at most two isomorphism classes of derived subgroups ( [formula omitted, refer to PDF] 2-groups) is investigated. A complete description of [formula omitted, refer to PDF] 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 [formula omitted, refer to PDF] 2-groups, especially those with finite rank, where algebraic number fields play an important role. Also, detailed structural information about insoluble [formula omitted, refer to PDF] 2-groups is found, and the locally free [formula omitted, refer to PDF] 2-groups are characterized. [PUBLICATION ABSTRACT] The structure of groups which have at most two isomorphism classes of derived subgroups ( sub(2)-groups) is investigated. A complete description of sub(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 sub(2)-groups, especially those with finite rank, where algebraic number fields play an important role. Also, detailed structural information about insoluble sub(2)-groups is found, and the locally free sub(2)-groups are characterized. 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. |
| Author | SMITH, HOWARD LONGOBARDI, PATRIZIA ROBINSON, DEREK J. S. MAJ, MERCEDE |
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| Cites_doi | 10.1007/BF01350807 10.1112/S0024610705006484 10.1016/0021-8693(72)90058-0 10.1093/acprof:oso/9780198507284.001.0001 10.1090/S0002-9947-1903-1500650-9 10.1016/0022-4049(75)90004-3 10.1007/s00013-005-1567-8 10.1090/conm/402/07578 |
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| References | Granville (S0017089512000821_ref4) 1988; 306 S0017089512000821_ref10 Robinson (S0017089512000821_ref9) 1976; 17 S0017089512000821_ref7 de Giovanni (S0017089512000821_ref2) 2006; 86 S0017089512000821_ref8 S0017089512000821_ref5 S0017089512000821_ref6 S0017089512000821_ref3 S0017089512000821_ref1 |
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| Snippet | The structure of groups which have at most two isomorphism classes of derived subgroups ($\mathfrak{D}$2-groups) is investigated. A complete description of... The structure of groups which have at most two isomorphism classes of derived subgroups ( $\mathfrak{D}$ 2 -groups) is investigated. A complete description of... Abstract The structure of groups which have at most two isomorphism classes of derived subgroups ( [formula omitted, refer to PDF] 2-groups) is investigated. A... The structure of groups which have at most two isomorphism classes of derived subgroups ( sub(2)-groups) is investigated. A complete description of... |
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| SubjectTerms | Algebra Algebraic group theory Isomorphism Mathematical analysis Number theory Subgroups |
| Title | ON GROUPS WITH TWO ISOMORPHISM CLASSES OF DERIVED SUBGROUPS |
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