Geodesic
Groups Satisfying Weak Chain Conditions on Certain Non-Normal Subgroups
Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 853-866
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In this paper, (generalized) soluble groups for which the set of all subgroups having infinite index in their normal closure satisfies some weak chain condition are described. 
@article{BUMI_2007_8_10B_3_a27,
     author = {De Mari, Fausto},
     title = {Groups {Satisfying} {Weak} {Chain} {Conditions} on {Certain} {Non-Normal} {Subgroups}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {853--866},
     year = {2007},
     volume = {Ser. 8, 10B},
     number = {3},
     zbl = {1184.20027},
     mrnumber = {2507901},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a27/}
}
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De Mari, Fausto. Groups Satisfying Weak Chain Conditions on Certain Non-Normal Subgroups. Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 853-866. http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a27/