The Schur property for subgroup lattices of groups |
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Authors: | Maria De Falco Francesco de Giovanni Carmela Musella |
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Institution: | (1) Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I - 80126 Napoli, Italy |
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Abstract: | A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup G′ of G is finite. A lattice analogue of this result is proved in this paper: if a group G contains a modularly embedded subgroup of finite index, then there exists a finite normal subgroup N of G such that G/N has modular subgroup lattice. Here a subgroup M of a group G is said to be modularly embedded in G if the lattice is modular for each element x of G. Some consequences of this theorem are also obtained; in particular, the behaviour of groups covered by finitely many subgroups
with modular subgroup lattice is described.
Received: 16 October 2007, Final version received: 22 February 2008 |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 20E15 20F14 |
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