Geodesic
Locally soluble-by-finite groups with small deviation for non-subnormal subgroups
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 1, pp. 1-7.

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A group $G$ has subnormal deviation at most $1$ if, for every descending chain $H_{0}>H_{1}>\dots $ of non-subnormal subgroups of $G$, for all but finitely many $i$ there is no infinite descending chain of non-subnormal subgroups of $G$ that contain $H_{i+1}$ and are contained in $H_{i}$. This property $\frak P$, say, was investigated in a previous paper by the authors, where soluble groups with $\frak P$ and locally nilpotent groups with $\frak P$ were effectively classified. The present article affirms a conjecture from that article by showing that locally soluble-by-finite groups with $\frak P$ are soluble-by-finite and are therefore classified.
Classification : 20E15, 20F14, 20F19
Keywords: subnormal subgroups; locally soluble-by-finite groups 
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Kurdachenko, Leonid A.; Smith, Howard. Locally soluble-by-finite groups with small deviation for non-subnormal subgroups. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 1, pp. 1-7. http://geodesic.mathdoc.fr/item/CMUC_2007__48_1_a0/