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Groups in which all subgroups are subnormal. (English) Zbl 0608.20017

The author shows for the class \(N_ 0\) of groups given in the title, that the derived series of every \(N_ 0\)-group terminates after a finite number of steps. Main steps in the proof are (i) a generalization of a theorem of C. J. B. Brookes [Bull. Lond. Math. Soc. 15, 235-238 (1983; Zbl 0506.20012)] leading to subgroups \(K\) and \(H\) such that every \(X\) with \(K\supseteq X\supseteq H\) is of defect at most some \(r\), (ii) the exhibition of a bound \(g(r)\) such that \(K^{(g(r))}\subseteq H\), depending on \(r\). Here \(g(r)=\sum^{r}_{i=1}f(i)\), where \(f(i)\) is the bound of the derived length of a group \(G\) having all subgroups subnormal of defect at most \(i\). J. E. Roseblade’s famous paper on this problem [J. Algebra 2, 402-412 (1965; Zbl 0135.04901)] mentions such a bound \(f(i)\).
Reviewer: H.Heineken

MSC:

20E15 Chains and lattices of subgroups, subnormal subgroups
20E26 Residual properties and generalizations; residually finite groups
20F14 Derived series, central series, and generalizations for groups
20F22 Other classes of groups defined by subgroup chains
20F19 Generalizations of solvable and nilpotent groups
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