A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup 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 L(<x,M>) 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.

The Schur property for subgroup lattices of groups / DE FALCO, Maria; DE GIOVANNI, Francesco; Musella, Carmela. - In: ARCHIV DER MATHEMATIK. - ISSN 0003-889X. - STAMPA. - 91:(2008), pp. 97-105.

The Schur property for subgroup lattices of groups

DE FALCO, MARIA;DE GIOVANNI, FRANCESCO;MUSELLA, CARMELA
2008

Abstract

A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup 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 L() 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.
2008
The Schur property for subgroup lattices of groups / DE FALCO, Maria; DE GIOVANNI, Francesco; Musella, Carmela. - In: ARCHIV DER MATHEMATIK. - ISSN 0003-889X. - STAMPA. - 91:(2008), pp. 97-105.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11588/315460
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