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Periodic groups with nearly modular subgroup lattice. (English) Zbl 1032.20021

A subgroup of a group \(G\) is called modular if it is a modular element of the lattice \({\mathcal L}(G)\) of all subgroups of \(G\). Lattices in which all elements are modular are also called modular. Thus the subgroup lattice of an Abelian group is modular and K. Iwasawa [J. Fac. Sci. Univ. Tokyo, Sect. I 4, 171-199 (1941; Zbl 0061.02503), Jap. J. Math. 18, 709-728 (1943; Zbl 0061.02504)] and R. Schmidt [Arch. Math. 46, No. 2, 118-124 (1986; Zbl 1027.20501)] have determined the structure of those groups with modular subgroup lattice.
By contrast a subgroup \(H\) of a group \(G\) is called ‘nearly modular’ if it has finite index in a modular subgroup of \(G\), and nearly modular elements of an arbitrary lattice can also be defined. A lattice is then called nearly modular if all its elements are nearly modular. The purpose of the paper under review is to obtain a result analogous to those mentioned earlier.
The main result is the following pleasing theorem: A periodic group \(G\) has nearly modular subgroup lattice if and only if there is a finite normal subgroup \(N\) of \(G\) such that the subgroup lattice \({\mathcal L}(G/N)\) is modular. This can be regarded as a lattice analogue of the theorem of B. H. Neumann [Math. Z. 63, 76-96 (1955; Zbl 0064.25201)] that a group \(G\) is finite-by-Abelian if and only if for each subgroup \(H\) of \(G\) the index \(|H^G:H|\) is finite.

MSC:

20E15 Chains and lattices of subgroups, subnormal subgroups
20F50 Periodic groups; locally finite groups
20E07 Subgroup theorems; subgroup growth
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