Finite \(p\)-groups in which the number of subgroups of possible order is less than or equal to \(p^3\). (Q606365)
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scientific article; zbMATH DE number 5816614
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Finite \(p\)-groups in which the number of subgroups of possible order is less than or equal to \(p^3\). |
scientific article; zbMATH DE number 5816614 |
Statements
Finite \(p\)-groups in which the number of subgroups of possible order is less than or equal to \(p^3\). (English)
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17 November 2010
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The \(p\)-groups with the title property are investigated. For \(p=2\) the results are not so full as for \(p\) odd. It appears, in particular, that if \(p>2\) and for all positive integers \(k\), the number of subgroups of order \(p^k\) in a \(p\)-group \(G\) is at most \(p^3\) (by Sylow, \(<p^3\)), then \(d(G)\leq 3\) (Theorem 2.5 presents the list of all such groups).
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inner Abelian \(p\)-groups
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metacyclic \(p\)-groups
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groups of order \(p^n\) with cyclic subgroups of index \(p^2\)
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numbers of subgroups
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0.9578316
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0.9311453
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0.89025575
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0.8893492
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0.8827391
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0.8802415
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