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Wielandt series and defects of subnormal subgroups in finite soluble groups. (English) Zbl 0794.20027

While R. A. Bryce and J. Cossey [J. Lond. Math. Soc., II. Ser. 40, No. 2, 244-256 (1989; Zbl 0734.20010)] have shown that there is a bound on the derived length of a soluble group in terms of its Wielandt length, a highlight of the present article is the following result (Corollary 2 of Theorem 2): Let \(wl(G)\), \(l(G)\) and \(b(G)\) denote the Wielandt length, the Fitting length and the maximum of the subnormal defects of a group \(G\), respectively. Then there exists a function \(f:\mathbb{N}\times\mathbb{N}\to\mathbb{N}\) such that every soluble group \(G\) satisfies the inequality \(wl(G)\leq f(l(G),b(G))\).

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D35 Subnormal subgroups of abstract finite groups
20D30 Series and lattices of subgroups
20E15 Chains and lattices of subgroups, subnormal subgroups
20F14 Derived series, central series, and generalizations for groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups

Citations:

Zbl 0734.20010
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References:

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