Generalized quasinormal subgroups of order \(p^2\) (Q2818396)
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scientific article; zbMATH DE number 6624875
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Generalized quasinormal subgroups of order \(p^2\) |
scientific article; zbMATH DE number 6624875 |
Statements
7 September 2016
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finite \(p\)-groups
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projectivities
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quasinormal subgroups
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Generalized quasinormal subgroups of order \(p^2\) (English)
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Let \(H\) be a subgroup of a group \(G\). If \(\langle H,K\rangle=HKHK\) for every cyclic subgroup \(K\) of \(G\) then \(H\) is said to be 4-quasinormal in \(G\). Let \(G=\langle H,K\rangle\) be a finite \(p\)-group (\(p\geq 5\)) where \(H\) is an elementary abelian subgroup of rank 2 and \(K\) is a cyclic subgroup. The author proves that in this case if \(H\) is 4-quasinormal in \(G\) then \(K^p\) is contained in the center of \(G\) and \(H\) is invariant under index-preserving projectivities.
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