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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Zbl MR
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. 
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/
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     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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