By a derived subgroup in a group G is meant the derived (or commutator) subgroup of a subgroup of G. It is natural to enquire how important the derived subgroups are within the lattice of all subgroups. Recently there has been interest in imposing restrictions on the number of derived subgroups in a group and investigating the resulting effect on the structure of the group. In this paper the structure of groups which have at most two isomorphism classes of derived subgroups (D2-groups) is investigated. A complete description of D2- 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 D2-groups, especially those with finite rank, where algebraic number fields play an important role. Also detailed structural information about insoluble D2-groups is found, and the locally free D2-groups are characterized. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM.
On Groups with Two Isomorphism Classes Of Derived Subgroups
LONGOBARDI, Patrizia;MAJ, Mercede;
2013-01-01
Abstract
By a derived subgroup in a group G is meant the derived (or commutator) subgroup of a subgroup of G. It is natural to enquire how important the derived subgroups are within the lattice of all subgroups. Recently there has been interest in imposing restrictions on the number of derived subgroups in a group and investigating the resulting effect on the structure of the group. In this paper the structure of groups which have at most two isomorphism classes of derived subgroups (D2-groups) is investigated. A complete description of D2- 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 D2-groups, especially those with finite rank, where algebraic number fields play an important role. Also detailed structural information about insoluble D2-groups is found, and the locally free D2-groups are characterized. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
