The structure of (generalized) soluble groups for which the set of all subnormal non-normal subgroups satisfies the maximal condition is described, taking as a model the known theory of groups in which normality is a transitive relation.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm103-1-10, author = {Fausto De Mari and Francesco de Giovanni}, title = {Groups satisfying the maximal condition on subnormal non-normal subgroups}, journal = {Colloquium Mathematicae}, volume = {103}, year = {2005}, pages = {85-98}, zbl = {1085.20013}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm103-1-10} }
Fausto De Mari; Francesco de Giovanni. Groups satisfying the maximal condition on subnormal non-normal subgroups. Colloquium Mathematicae, Tome 103 (2005) pp. 85-98. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm103-1-10/