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/