×

zbMATH — the first resource for mathematics

A lattice characterization of the lattice of subgroups of an Abelian group containing two independent aperiodic elements. (Italian) Zbl 0595.20025
An element, x, of a lattice L is cyclic if x/0 is distributive and satisfies the maximal condition, and x is torsion-free if x/0 is free of atoms. The author defines L to be a G-lattice if it satisfies conditions Cl-C8 (which are too detailed to give here). He continues work from an earlier paper by proving that L is isomorphic with the lattice of subgroups of an abelian group which contains two independent aperiodic elements if and only if L is a G-lattice which contains nonzero cyclic torsion-free elements x and y such that \(x\wedge y=0\).
Reviewer: W.E.Deskins
MSC:
20E15 Chains and lattices of subgroups, subnormal subgroups
20K27 Subgroups of abelian groups
06B15 Representation theory of lattices
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] S.A. Anishchenko , Representations of modular lattices by lattices of subgroups , Mat. Zap. Krasnoyarskogo Gos. Ped. In-ta , 1 ( 1965 ), pp. 1 - 21 .
[2] R. Baer , The significance of the system of subgroups for the structure of the group , Amer. J. of Math. , 61 ( 1939 ), pp. 1 - 44 . MR 1507945 | JFM 65.0060.01 · Zbl 0020.34704 · doi:10.2307/2371383
[3] H. Ribeiro , Lattices des groupes abeliens finis , Comm. Math. Helv. , 23 ( 1949 ), pp. 1 - 17 . MR 30517 | Zbl 0037.15702 · Zbl 0037.15702 · doi:10.1007/BF02565588 · eudml:138972
[4] L.E. Sadovskii , Projectivities and isomorphisms of nilpotent groups , Izv. Akad. Nauk SSSR ser. Mat. , 29 ( 1965 ), pp. 171 - 208 . MR 171848
[5] C.M. Scoppola , Sul reticolo dei sottogruppi di un gruppo abeliano senza torsione di rango diverso da 1: una caratterizzazione reticolare , Rend. Sem. Mat. Univ. Padova , 65 ( 1981 ), pp. 205 - 221 . Numdam · numdam:RSMUP_1981__65__205_0
[6] B.V. Yacovlev , Conditions under which a lattice is isomorphic to a lattice of subgroups of a groups , Algebra i Logika , 13 ( 1974 ), pp. 694 - 712 . MR 401933 | Zbl 0438.20019 · Zbl 0438.20019 · doi:10.1007/BF01462952 · eudml:186866
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.