On the join of subnormal elements in a lattice

Robert Leroy Kruse

Vol. 28 (1969), No. 3, 571–574

DOI: 10.2140/pjm.1969.28.571

Abstract

Of fundamental importance to the study of subnormal subgroups is the following result of Wielandt:

Let A and B be subnormal subgroups of a group G such that A is normal in A ∪ B. Then A ∪ B is subnormal in G.

The usual proof of Wielandt’s result depends on the construction by conjugation of a special subnormal series from A to G. It would be of interest to obtain a proof which uses only the given subnormal series, without explicit dependence on conjugation, and valid in algebraic systems other than groups.

This note presents, in the more general context of a lattice with the normality relation introduced by R. A. Dean, a proof of the analogous result in case either A or B has defect three or less.

Mathematical Subject Classification

Primary: 06.30

Milestones

Received: 19 March 1968

Published: 1 March 1969