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On the functorial method for studying the lattices of subgroups of finite groups. (Russian, English) Zbl 0983.20018

Sib. Mat. Zh. 42, No. 1, 30-40 (2001); translation in Sib. Math. J. 42, No. 1, 25-33 (2001).
A functorial approach is proposed for generalizing H. Wielandt’s result [Math. Z. 45, 209-244 (1939; Zbl 0021.21003)] that all subnormal subgroups of a group form a lattice. Axiomatizing the main properties of subnormal subgroups (invariance under homomorphisms, transitivity, heredity in subgroups), the authors introduce the notion of natural, transitive, lattice functor and describe all lattices induced by such functors in finite solvable groups. As a consequence, the hereditary formations with the lattice property in the universe of all finite solvable groups are described. Some open problems are formulated.

MSC:

20D30 Series and lattices of subgroups
20D35 Subnormal subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks

Citations:

Zbl 0021.21003
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