## Leonid A. Kurdachenko, Howard Smith

*Locally soluble-by-finite groups with small deviation for non-subnormal subgroups *

Comment.Math.Univ.Carolin. 48,1 (2007) 1-7. **Abstract:**A group $G$ has subnormal deviation at most $1$ if, for every descending chain $H_{0}>H_{1}>...$ 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.

**Keywords:** subnormal subgroups, locally soluble-by-finite groups

**AMS Subject Classification:** 20E15, 20F19

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