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On Groups that are Isomorphic with Every Subgroup of Finite Index and their Topology
Author(s) -
Robinson Derek J. S.,
Timm Mathew
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005766
Subject(s) - commutator subgroup , mathematics , centralizer and normalizer , abelian group , omega and agemo subgroup , pure mathematics , homology (biology) , characteristic subgroup , fitting subgroup , group (periodic table) , normal subgroup , torsion subgroup , combinatorics , discrete mathematics , topology (electrical circuits) , elementary abelian group , locally finite group , physics , biochemistry , chemistry , quantum mechanics , gene
The main result is that a finitely generated group that is isomorphic to all of its finite index subgroups has free Abelian first homology, and that its commutator subgroup is a perfect group. A number of corollaries on the structure of such groups are obtained, including a method of constructing all such groups for which the commutator subgroup has a trivial centralizer. As an application, conditions are presented for the covering spaces of compact manifolds that determine when the fundamental groups of the base spaces are free Abelian.