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- Title
The Schur property for subgroup lattices of groups.
- Authors
Maria De Falco; Francesco de Giovanni; Carmela Musella
- Abstract
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 $${\mathfrak{L}}(\langle x,M \rangle)$$ 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.
- Subjects
MODULES (Algebra); ALGEBRA; RING theory; FINITE groups
- Publication
Archiv der Mathematik, 2008, Vol 91, Issue 2, p97
- ISSN
0003-889X
- Publication type
Article
- DOI
10.1007/s00013-008-2624-x