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A note on subnormal defect in finite soluble groups. (English) Zbl 0673.20011

All groups discussed are finite and soluble. For each positive integer \(n\), denote by \(B_ n\) the class of groups whose subnormal subgroups have defect at most \(n\). Casolo has proved that \(B_ 2\)-groups have derived length at most 5, and Hawkes showed that there is no bound on the derived length of \(B_ 3\)-groups. The author gives a neat construction for \(B_ 3\)-groups of arbitrary derived length having abelian Sylow subgroups. Hawkes’ examples do not have this latter property, that is, are not A-groups.
Reviewer: D.J.McCaughan

MSC:

20D35 Subnormal subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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References:

[1] Zacher, Ricerche Mat. 1 pp 287– (1952)
[2] DOI: 10.1007/BF01235317 · Zbl 0418.20018 · doi:10.1007/BF01235317
[3] DOI: 10.1007/BF01196648 · Zbl 0547.20017 · doi:10.1007/BF01196648
[4] Casolo, Rend. Sem. Mat. Univ. Padova 71 pp 257– (1984)
[5] Cossey, Groups of odd order in which every subnormal subgroup has defect at most two (1987) · Zbl 0759.20007
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