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Finite soluble groups in which all subnormal subgroups have defect at most 2. (Italian) Zbl 0575.20019

It is proved that if every subnormal subgroup of the finite soluble group G is of subnormal defect at most 2, then the Fitting length of G is at most 4 and the derived length at most 5. The holomorph of the elementary Abelian group of order 9 shows that these bounds are best possible. Among the results used are theorems of Heineken, who proved that the class of nilpotent groups of this type is at most 4, and McCaughan and Stonehewer, who had previously obtained the bound 9 for the Fitting length. Among the lemmas used is the assertion (Lemma 2) that if a non-Abelian group G operates faithfully on an elementary Abelian group \(V=V_ 1\times V_ 2\) and G is irreducible on each \(V_ i\) then the G-module V is generated by a single element. It is easy to reduce this to the case when G operates irreducibly on U and \(Aut_ GU\) is transitive on the non-zero elements of U; but then \(End_ GU\) is field over which the dimension of U is 1, so \(End_ GU\) is self-centralizing on U and the image of G is Abelian. The bound on the Fitting length follows from the more general assertion (Theorem 3) that \(G/F_ 2(G)\) is supersoluble and metabelian.
Reviewer: N.Blackburn

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D35 Subnormal subgroups of abstract finite groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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