On the functorial method for studying the lattices of subgroups of finite groups (Q2760741)
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scientific article; zbMATH DE number 1682269
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the functorial method for studying the lattices of subgroups of finite groups |
scientific article; zbMATH DE number 1682269 |
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13 December 2001
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subnormal subgroups
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lattices of subgroups
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lattice functors
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hereditary formations
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finite solvable groups
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0.91649175
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0.9154811
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0.9125655
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On the functorial method for studying the lattices of subgroups of finite groups (English)
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A functorial approach is proposed for generalizing \textit{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.
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