On strong uniform dimension of finite groups.
by
Agnieszka Sakowicz
University of Bialystok, Poland
Coauthors: Czeslaw Baginski (University of Bialystok), Jan Krempa (Warsaw University)

Let L be a lattice with 0. We will say that L is balanced (permutable in [1, 2]) if for all x, y, z from L we have

x /\ y=0   &  ( x \/ y) /\ z=0  ===>  (y \/ z) /\ x=0   &  (z \/ x) /\ y=0

and L is strongly balanced (globally permutable in [1, 2]) if all nonempty intervals of L are balanced. Clearly any sublattice with 0 of a balanced lattice is balanced and any sublattice of a strongly balanced lattice is strongly balanced. Furthermore modular lattices are always strongly balanced, but not conversly.

The uniform dimension was introduced by Goldie as an invariant of modules. The notions of balanced and strongly balanced lattices were introduced and studied in [2, 3]. In particular, in this paper it is proved that the uniform and the strong uniform dimensions can be well defined only in some of such lattices.

In [1] all finite groups with strongly balanced lattices of subgroups were described. In this talk we show how to calculate the strong uniform dimension for all strongly balanced lattices of subgroups of finite groups.

References

[1] C. Baginski, A. Sakowicz, Finite groups with globally permutable lattices of subgroups. Colloq. Math. 82(1999), no.1, 65-77.

[2] J. Krempa, B. Terlikowska-Osowska, On uniform dimension of lattices, Contributions to General Algebra 9, Wien 1995 - Verlag B. G. Teubner, Stuttgart.

[3] A.P. Zolotarev, Balanced lattices and and Goldie numbers in balanced lattices, Sibirsk. Mat. Zh. 35(1994), 602-611.

Date received: June 29, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-24.